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JAN  22  1942 


2  1954 

APR  2  2  RECfl 


On  the 

Foundation  and  Technic 
of  Arithmetic 


By 

George  Bruce  Halsted 

A.  B.  and  A.  M.,  Princeton;  Ph.  D.,  Johns  Hopkins;  F.  R.  A.  S. 
2-   S  S  O  O 


Chicago 

The  Open  Court  Publishing  Company 
1912 


COPYRIGHT  BY 

THE  OPEN  COURT  PUBLISHING  CO. 
1912 


J-ihra 


CONTENTS. 

CHAPTER  PACK 

Introduction I 

I.  The  Prehuman  Contributions  to  Arithmetic  3 

The  natural  individual,  3. — The  artificial  individual,  4. — Primary 
number,  5. — Our  base  ten,  7. 

II.  The  Genesis  of  Number 8 

III.  Counting  and  Numerals  10 

Correlation,  10. — To  count,  n. — The  primitive  standard  sets,  n. 
— The  abacus,  12. — The  word-numeral  system,  12. — Periodicity,  13. 
— A  partitioned  unit,  14. — Number  without  counting,  14. — Decimal 
word-numerals,  14. — Invariance  of  cardinal,  15. 

IV.  Genesis  of  our  Number  Notation  17 

Positional  counting,  17. — The  abacus,  17. — Recorded  symbols,  18. 
—The  Hindu  numerals,  19. — The  zero,  20. — Our  present  notation, 

22. 

V.  The  Two  Direct  Operations,  Addition  and  Multiplication.     26 

Notation,  26. — The  symbol  =,  26. — Inequality,  27. — Parentheses, 
28. — Expressions,  28. — Substitution,  29. — Addition,  29. — Formulas, 
32. — Ordinal  addition,  33. — Properties  of  addition,  33. — Multiplica- 
tion, 35. 

VI.  The  Two  Inverse  Operations,  Subtraction  and  Division. .    39 

Inversion,  39. — Subtraction,  39. — Division,  41. 
VII.  Technic  44 

Addition,  44.— Subtraction,  44. — Multiplication,  45. — Verify  multi- 
plication, 46. — Shorter  forms,  47. — Division,  47. — Verify  division, 
48. 

VIII.  Decimals    49 

Decimals,  49. — Product,  52. — Quotient,  53. 

IX.  Fractions 55 

Generalizations'  of  number,  55. — Principle  of  permanence,  56. — 
Fractions,  56. — Fractions  ordered,  59. — Division  of  fractions,  61. — 
Multiplication  of  fractions,  61. 

X.  Relation  of  Decimals  to  Fractions 63 

ist  Decimals  into  fractions,  63. — 2d.  Fractions  into  decimals,  63. 
—Base,  65.— Change  of  base,  67. 


IV  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

CHAPTER  PAGE 

XI.  Measurement 68 

Why  count  ?  68. — The  measure  device,  70. — Counting  prior  to  meas- 
uring, 70. — New  assumptions,  72. 

XII.  Mensuration  75 

Geometry,  75. — Length  of  a  sect.  76. — Length  of  the  circle,  76. — 
Area,  77. — Volume,  78. 

XIII.  Order 81 

Depiction,  82. — Infinite,  82. — Sense,  82. — Analysis  of  order,  83. — 
Ordered  set,  84. — Finite  ordinal  types,  85. — Number  series,  type  of 
order,  85. — Well-ordered  sets,  86. 

XIV.  Ordinaf  Number  88 

Ordinal  number,  88. — Children's  counting,  88. — Uses  of  ordinals, 
89. — Nominal  number,  92. 

XV.  The  Psychology  of  Reading  a  Number 94 

XVI.  Arithmetic  as  Formal  Calculus  101 

XVII.  On  the  Presentation  of  Arithmetic no 

ist  Grade:  Previous  blunders,  no.  —  Begin  with  ordinals,  no. 
— Cardinal  from  ordinal,  in.  —  Ordinal  counting,  in.  —  Cardinal 
counting,  112. — Number  precedes  measure,  112. — Cardinal  number, 
113. — How  to  begin,  113. — Ordinal  games,  114. — The  call,  114. — 
Ordinal  operations,  116. — The  simplest  cardinal,  116. — Triplets  and 
quartets,  117. — The  "how  many"  idea,  117. — Symbols,  117. — Car- 
dinal counting,  117. — Recognition  of  the  cardinal,  117. — Cardinal 
addition,  118. — Summary,  118. — 2d  Grade:  Measurement,  119. — 
The  decimal,  120. — Carrying,  121. — Subtraction,  121. — Fractions, 
122. —  Multiplication,  122.  —  Division,  123.  —  Summary,  124.  —  3d 
Grade,  125. — 4th  Grade,  127. — sth  Grade,  127. — 6th  Grade,  128. — 
7th  Grade,  129. 

Index 131 


INTRODUCTION. 

In  the  French  Revolution,  when  called  before  the 
tribunal  and  asked  what  useful  thing  he  could  do  to 
deserve  life,  Lagrange  answered:  "I  will  teach  arith- 
metic." 

Almost  invariably  now  arithmetic  is  taught  by  those 
whose  knowledge  of  mathematics  is  most  meager.  No 
wonder  it  and  the  children  suffer.  In  this  day  of  the 
arithmetization  of  mathematics  and  later  its  logiciza- 
tion,  are  the  beauty,  the  elegance  of  arithmetical  proce- 
dures to  remain  still  unexplained?  Is  the  singular,  the 
lonely  precision  of  this  science  and  art  to  remain  un- 
heralded, unexpounded? 

In  arithmetic  a  child  may  taste  the  joy  of  the  genius, 
the  joy  of  creative  activity. 

Arithmetic  is  for  man  an  integrant  part  of  his  world 
construction.  Thus  do  his  fellows  make  their  world, 
and  so  must  he.  Now  this  is  not  by  passive  apprehension 
of  something  presenting  itself,  but  by  permeating  vitali- 
zation  spreading  life  and  its  substance  through  what  the 
ignorant  teacher  would  present  as  the  dead  mechanism 
of  mechanical  computation. 

More  than  in  any  other  science,  there  has  been  in 
mathematics  an  outburst  of  most  unexpected,  most  deep- 
reaching  progress.  Its  results,  if  made  available  for  the 


2  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

teacher,  will  revivify  this  first,  most  precious  of  edu- 
cational organisms;  the  more  so  since  mathematics  is 
seen  to  possess  of  all  things  the  most  essential,  most 
fundamental  objective  reality. 


CHAPTER  I. 

THE  PREHUMAN  CONTRIBUTIONS  TO  ARITH- 
METIC. 

Properly  to  understand  or  to  teach  arithmetic,  one 
should  have  a  glimpse  of  its  origin,  foundation,  meaning, 
aim. 

Arithmetic  is  the  science  of  number,  but  for  the  ordi- 
nary school-teacher  it  is  to  be  chiefly  t|ie  doctrine  of  pri- 
mary natural  number,  the  decimal  and  later  the  fraction, 
and  the  art  of  reckoning  with  them. 

Numbers  are  of  human  make,  creations  of  man's 
mind;  but  they  are  first  created  upon  and  influenced  by 
a  basis  which  comes  from  the  prehuman. 

Before  our  ancestors  were  men,  they  represented  to 
themselves,  as  do  some  animals  now,  the  world  as  con- 
The  natural  sisting  of  or  containing  individuals,  definite 
individual.  objects  of  thought,  things.  They  exercised 
an  individuating  creative  power.  In  now  understanding 
by  thing  a.  definite  object  of  thought,  conceived  as  indi- 
vidual, we  are  using  a  method  of  world  presentation 
which  served  animals  before  there  were  any  men  to  serve. 

The  child's  consciousness  certainly  begins  with  a 
sense-blur  into  which  specification  is  only  gradually  in- 
troduced. At  what  stage  of  animal  development  the 
vague  and  fluctuating  fusion,  which  was  the  world,  be- 
gins to  be  broken  up  into  persistently  separate  entities 


4  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

would  be  an  interesting  comparative  biologi co-psycho- 
logic investigation.  However  that  might  turn  out,  yet 
things,  separate  objective  things,  are  a  gift  to  man  from 
the  prehuman.  Yet  simple  multiplicity  of  objects  present 
to  perception  or  even  to  consciousness  does  not  give 
number.  The  duck  does  not  count  its  young.  The  crow, 
wise  old  bird,  has  no  real  counting  power  to  help  its 
cunning.  The  animals'  senses  may  be  keener  than  ours, 
yet  they  never  give  number. 

A  babe  sees  nothing  numeric.  Even  an  older  child 
may  attend  to  diverse  objects  with  no  suggestion  from 
them  of  number.  Sense-perception  may  be  said  to  have 
to  do  with  natural  individuals,  but  never,  unaided  by 
other  mind-act,  does  it  give  number. 

To  the  animal  habit  of  postulating  entities  as  separate 
must  be  added,  before  cardinal  number  comes,  the  human 
The  artificial  unification  of  certain  of  them  into  one  whole, 
individual.  one  totality,  one  assemblage  or  group  or  set, 
one  discrete  aggregate  or  artificial  individual  man-made. 

This  artificial  whole,  this  discrete  aggregate  it  is  to 
which  cardinal  number  pertains.  Thus  number  rests 
upon  a  prehuman  basis,  yet  is  not  number  itself  pre- 
human. Cardinal  number  involves  more  than  the  animal 
or  natural  individuals  or  things.  It  comes  only  with  a 
human  creation,  the  creation  of  artificial  individuals,  dis- 
crete aggregates  taken  each  as  an  individual,  an  indi- 
vidual of  human  make,  fleeting  perhaps  as  our  thought, 
transient,  yet  the  necessary  substratum  for  cardinal  num- 
ber. Unification  is  necessary.  The  mind  must  make  of 
the  distinct  things  a  whole,  a  totality.  Else  no  cardinal 
number. 

Now  to  an  educated  man  a  number  concept  is  sug- 
gested when  a  specific  simple  aggregate  of  objects  is  at- 


PREHUMAN  CONTRIBUTIONS  TO  ARITHMETIC.  5 

tended  to.  Not  so  to  any  animal,  though  just  the  same 
individual  objects  be  recognized  and  attended  to.  The 
animal  has  the  unity  of  the  natural  object  or  individual, 
but  that  unity  is  not  enough.  There  is  needed  the  new, 
the  artificial,  the  man-made  individuality  of  the  total 
aggregate.  To  this  artificial  individual  it  is  that  the 
cardinal  number  pertains.  There  is  thus  a  unity,  man- 
made,  of  the  aggregate  of  natural  individuals,  of  the  set 
of  constituent  units.  To  this  unity  made  of  units  car- 
dinal number  belongs. 

Going  for  quite  different  articles,  or  to  accomplish 
entirely  different  things,  may  we  not  help  and  check 
memory  by  fixing  in  our  mind  that  we  are  to  get  three 
things,  or  that  we  are  to  do  three  things?  How  man- 
made,  arbitrary,  and  artificial,  this  conjoining  of  acts 
most  diverse  into  a  fleeting  unified  whole! 

Each  finger  of  the  left  hand  is  different.  A  dog 
might  be  taught  to  recognize  each  as  a  separate  and  dis- 
tinct individual.  Only  a  man  can  make  of  all  at  once  an 
individual  which,  conceived  as  a  whole,  is  yet  multiple, 
multiplex,  a  manifold,  fivefold,  a  five  of  fingers,  a  prod- 
uct of  rational  creation  beyond  the  dog. 

A  primary  cardinal  number  is  a  character  or  attribute 
of  an  artificial  unit  made  of  natural  units.  It  needs  this 
Primary  single  individuality  and  this  multiplicity  of 

number.  individuals.  The  fingered  hand  has  fiveness 

only  if  taken  as  an  individual  made  of  individuals. 

Number  is  a  quality  of  a  construct.  If  three  things 
are  completely  amalgamated,  emulsified,  like  the  com- 
ponents of  bronze  or  the  ingredients  of  a  cake,  there  re- 
mains no  threeness.  If  some  things  are  in  no  way  taken 
together  the  number  concept  is  still  inapplicable,  we  do 
not  see  them  as  a  trio. 


6  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

The  animally  originated  primitive  individuals,  how- 
ever complete  in  their  distinctness,  have  no  numeric  sug- 
gestion. The  creative  synthesis  of  a  manifold  must  pre- 
cede the  conscious  perception  of  its  numeric  quality.  The 
set  must  be  conceived  as  a  whole  before  discriminated 
as  a  dozen.  It  is  only  to  man-made  conceptual  unities 
that  the  numeric  quality  pertains.  This  "number  of 
natural  individuals"  in  an  artificial  individual  is  called 
its  cardinal  number  or  cardinal.  The  cardinal  n  of  a  set 
j  is  the  class  of  all  sets  similar  to  s. 

Primary  number  would  seem  in  some  sense  a  normal 
creation  of  man's  mind.  No  primitive  language  has  ever 
been  investigated  without  therein  finding  records  of  the 
number  idea,  unmistakable  though  perhaps  slight,  limited, 
meager,  it  may  be  not  going  beyond  our  baby  stage,  one, 
two,  many. 

There  is  a  baby  stage  when  no  many  is  specialized 
but  two.  One,  two,  many,  then  baby  waits  how  long  be- 
fore that  many  called  three  is  specialized  ?  Numeric  one 
as  cardinal  only  comes  into  existence  in  contrast  with 
many.  It  involves  a  distinction  between  the  class  whose 
only  member  is  x,  and  the  thing  x  itself.  The  Stoic 
Chrysippos  (282-209  B.  C.)  spoke  of  the  "aggregate  or 
assemblage  one."  Number  comes  when  we  make  a  vague 
many  specific. 

The  world-mind  rose  from  the  animal  to  the  human 
when  it  grouped,  aggregated,  made  wholes  of,  made  arti- 
ficial individuals  of  the  distinct  individual  objects  pre- 
viously created  by  the  animal  mind.  We  may  see  babies 
recapitulating  the  race  in  this. 

The  number  of  a  particular  totality  represents  the  par- 
ticular multiplicity  of  its  individual  elements  and  nothing 
more.  So  far  as  represented  in  a  number,  each  natural 


PREHUMAN  CONTRIBUTIONS  TO  ARITHMETIC.  7 

individual  loses  everything  but  its  distinctness;  all  are 
alike,  indistinguishably  equivalent.  The  idea  of  unity 
is  doubly  involved  in  number,  which  applies  to  a  unity 
of  a  plurality  of  units.  The  units  are  arithmetically 
identical;  not  so  the  complex  unities  man-made  out  of 
collections  of  the  units.  To  these  pertain  the  differing 
cardinal  numbers. 

In  our  developed  number  systems  certain  manys  take 
Our  base  on  a  peculiar  prominence,  are  of  basal  char- 
acter. Of  these  ten  has  now  permanently 
the  upper  hand. 

What  is  the  origin  of  this  preeminence? 

Its  origin  is  prehuman.  Our  system  is  decimal,  not 
because  ten  is  scientifically,  arithmetically  a  good  base, 
a  superior  number,  but  solely  because  our  prehuman  an- 
cestors gave  us  five  fingers  on  each  of  two  hands. 


CHAPTER  II. 

THE  GENESIS  OF  NUMBER. 

In  nature,  distinct  things  are  made  and  perceived  as 
individual.     Each  distinct  thing  is  a  whole  by  itself, 

-     ..    .          a  qualitative  whole.     The  individual  thing 
Cardinals.         .      ,  ...  ,  . 

is  the  only  whole  or  distinct  object  in  na- 
ture. But  the  human  mind  takes  individuals  together 
and  makes  of  them  a  single  whole  of  a  new  kind,  and 
names  it.  Thus  we  have  made  the  concept  a  flock,  a 
herd,  a  bevy,  a  covey,  a  genus,  a  species,  a  bunch,  a  gang, 
a  host,  a  class,  a  family,  a  group,  an  array,  a  crowd,  a 
party,  an  assemblage,  an  aggregate,  a  manifold,  a  throw, 
a  set,  etc.  These  are  artificial  units,  discrete  magnitudes  ; 
the  unity  is  wholly  in  the  concept,  not  in  nature;  it  is 
artificial.  We  constitute  of  certain  things  an  artificial 
individual  when  we  distinguish  them  collectively  from 
the  rest  of  the  world,  making  out  of  subsidiary  individ- 
uals a  single  thing,  a  system,  of  which  each  component 
is  recognizable  as  distinct  from  all  others.  From  the 
contemplation  of  the  natural  individual  or  element  in 
relation  to  the  artificial  individual,  the  group,  spring  the 
related  ideas  "many"  and  "one."  We  must  have  numeric 
many  before  we  can  have  cardinal  one.  A  natural  quali- 
tative unit  thought  of  in  contrast  to  a  "many"  as  not- 
many  gives  the  idea  "one"  as  cardinal.  An  aggregate 
may  contain  only  a  single  element.  Thus  we  have  a  set 
containing  an  element  with  which  every  element  is  iden- 


THE  GENESIS  OF  NUMBER.  ^ 

tical.  So  we  get  "one."  A  unity,  a  "many"  composed 
of  a  "one"  and  another  "one"  is  characterized  as  two. 

The  unity,  the  "many"  composed  of  "one"  and  the 
special  many  "two"  is  characterized  as  three. 

Among  the  primitive  ideas  of  cardinal  number,  the 
idea  of  "two"  is  the  first  to  be  formed  definitely.  There 
are  ever  present  doublets,  things  which  can  be  grasped 
in  pairs.  This  two  is  the  very  simplest  many,  the  simplest 
recognized  form  of  plurality.  It  is  incalculably  simpler 
than  three,  as  witness  whole  savage  tribes  whose  spoken 
number  system  is  "one,  two,  many" ;  as  witness  the  mind- 
wasting  primitive  stupidities  of  the  dual  number  in  Greek 
grammar. 

The  special  many,  a  one  made  of  three,  a  trinity,  a 
trio,  triplets,  here  is  an  advance.  When  to  the  grasp  of 
the  pair,  the  dominance  over  the  trio  is  added,  when  the 
three  is  created,  then  after-progress  is  rapid. 

With  a  couple  of  pairs  goes  four;  with  a  couple  of 
threes,  six.  A  hand  represents  five  coming  in  between 
four  and  six.  A  pair  of  hands  says  ten.  A  pair  of  tens 
is  twenty,  a  score.  A  pair  of  fours  is  eight.  A  trio  of 
threes  is  nine.  A  pair  of  sixes  or  a  trio  of  fours  is  twelve, 
a  dozen. 

Arithmetic  flowers  like  a  rocket.  That  seven  is  left 
out,  is  missed,  makes  it  the  sacred,  the  mystic  number  of 
superstition.  To  numbers,  however  complicated  their 
genesis,  is  finally  ascribed  a  certain  objective  reality.  In 
our  mind  the  number  concepts  finally  become  simple 
things,  objectively  real. 


CHAPTER  III. 
COUNTING  AND  NUMERALS. 

The  ability  of  mind  to  relate  things  to  things,  to 

correlate,  to  represent  something  by  some- 
Correlation. 

thing  else,  to  make  or  perceive  a  correspon- 
dence between  things  or  thought  creations  is  funda- 
mental, essential,  necessary. 

The  operation  of  establishing  such  a  correspondence 
between  two  sets  that  every  thing  or  element  of  each 
set  is  mated  with,  paired  with,  just  one  particular  thing 
or  element  of  the  other,  is  called  establishing  a  one-to-one 
correspondence  between  the  sets.  Two  sets  which  can 
be  so  mated  are  said  to  be  equivalent  as  regards  plural- 
ity, or  to  have  the  same  potency.  Two  sets  equivalent 
to  the  same  are  equivalent  to  each  other,  their  elements 
correlated  to  the  same  element  being  thereby  mated. 
Two  sets  between  which  a  one-to-one  relation  exists  have 
the  same  cardinal  number  and  are  said  to  be  cardinally 
similar. 

A  set's  cardinal  number  is  what  is  common  to  the 
set  and  every  equivalent  set.  Thus  a  set's  cardinal  is 
independent  of  every  characteristic  or  quality  of  any 
element  beyond  its  distinctness.  To  find  the  cardinal  of 
a  set,  we  count  the  set. 

Counting  is  the  establishing  of  a  one-to-one  corres- 
pondence of  aggregates,  one  of  which  belongs  to  a  well- 
known  series  of  aggregates.  If  a  group  of  things  have 


COUNTING  AND  NUMERALS.  11 

this  correspondence  with  this  standard  group,  then  those 
properties  of  this  standard  group  which  are  carried  over 
by  the  correspondence  will  belong  to  the  new  group. 
They  are  properties  of  the  group's  cardinal  number. 

To  count  an  aggregate,  an  artificial  individual,  is  to 
identify  it  as  to  numeric  quality  with  a  familiar  assem- 
blage by  setting  up  a  one-to-one  correspon- 
To  count.  ,  ,    Al_ 

dence   between  the  elements  of    the    two 

groups.  Thus  counting  consists  in  assigning  to  each 
natural  individual  of  an  aggregate  one  distinct  individual 
in  a  familiar  set,  originally  a  group  of  fingers,  now  usu- 
ally a  set  of  words  or  marks.  So  counting  is  essentially 
the  numeric  identification,  by  setting  up  a  one-to-one  cor- 
respondence, of  an  unfamiliar  with  a  familiar  group. 
Thus  it  ascertains,  it  fixes  the  nature  of  the  less  familiar 
through  the  preceding  knowledge  of  the  more  familiar. 

Primitively  the  known  groups  were  the  groups  of 
fingers.  The  fingers  gave  the  first  set  of  standard  groups 
The  primitive  and  formed  the  original  apparatus  for  count- 
standard  sets.  mg)  ancj  serve(j  for  the  symbolic  transmis- 
sion of  the  concepts,  the  number  ideas  generated.  More 
than  that,  this  finger  counting  gave  the  names  of  the 
numbers,  the  numeric  words  so  helpful  in  the  further 
development  of  numeric  creation.  The  name  of  a  number, 
when  referring  to  an  artificial  unit,  as  of  sheep,  denoted 
that  a  certain  group  of  fingers  would  touch  successively 
the  natural  units  in  the  discrete  magnitude  indicated,  or 
a  certain  finger  would  stand  as  a  symbol  for  the  numerical 
characteristic  of  that  group  of  natural  units. 

Our  word  "five"  is  cognate  with  the  Latin  quinque, 
Greek  pente,  Sanskrit  pancha,  Persian  pendji;  now  in 
Persian  penjeh  or  pentcha  means  an  outspread  hand. 


12          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

In  Eskimo  "hand  me"  is  tamuche;  "shake  hands"  is 
tallalue;  "bracelet"  is  talegowruk;  "five"  is  talema. 

In  the  language  of  the  Tamanocs  of  the  Orinoco, 
five  means  "whole  hand" ;  six  is  "one  of  the  other  hand" ; 
and  so  up  to  ten  or  "both  hands." 

Philology  confirms  that  the  original  counting  series 
or  outfit  was  the  series  of  sets  of  fingers,  and  this  primi- 
tive method  preceded  the  formation  of  numeral  words. 
The  use  of  visible  signs  to  represent  numbers  and  aid 
reckoning  is  not  only  older  than  writing,  but  older  than 
the  development  of  numerical  language.  In  very  many 
languages  the  counting  words  come  directly  and  recog- 
nizably from  the  finger  procedure. 

But  of  the  fingers  there  are  only  a  few  distinct  ag- 
gregates, only  ten.  Developing  man  needs  more,  needs 
to  enlarge  and  extend  his  standards. 

The  Chinese,  even  at  the  present  day,   extend  the 
series  of  primary  groups,  the  finger-groups,  by  substi- 
tuting groups  of  counters  movably  strung 
The  abacus.  &  s       F     .  r  w*i 

on  rods  fixed  in  an  oblong  frame.     With 

this  abacus,  which  they  call  shwanpan,  reckoning  board, 
and  the  Japanese  call  soroban,  they  count  and  perform 
their  arithmetical  calculations. 

In  many  languages  there  are  not  even  words  for  the 
first  ten  groups.  Higher  races  have  not  only  named 
The  word-  *nese  grouPs>  but  have  extended  indefinitely 
numeral  this  system  of  names.  They  no  longer  count 
directly  with  their  fingers,  but  use  a  series 
of  names,  so  that  the  operation  of  counting  an  assemblage 
of  things  consists  in  assigning  to  each  of  them  one  of 
these  numeral  words,  the  words  being  always  taken  in 
order,  and  none  skipped,  each  word  being  thus  capable 
of  representing  not  merely  the  individual  with  which  it 


COUNTING  AND  NUMERALS.  13 

is  associated,  but  the  entire  named  group  of  which  this 
individual  is  the  last  named. 

In  making  this  series  of  word-numerals,  there  is 
evidently  need  for  a  system  of  periodic  repetition.  The 

prehuman  fixes  five,  ten,  or  twenty  as  the 

Periodicity.  r  ,  .  ,  .  .        ,  ^ r 

number  after  which  repetition  begins.     Ut 

these,  ten  has  become  predominant.  Thus  come  our 
word-numerals,  each  applicable  to  just  one  of  a  counted 
set  and  to  the  aggregate  ending  with  this  one.  This 
dekadic  word-system  makes  easy,  with  a  simple,  a  light 
numerational  equipment,  the  perfectly  definite  expression 
of  any  number,  however  advanced. 

So  for  us  to  count  is  to  assign  the  numerals  one,  two, 
three,  etc.,  successively  and  in  order,  to  all  the  individual 
objects  of  a  collection,  one  to  each.  The  collection  is 
said  to  be  given  in  number,  the  number  of  things  in  it, 
by  the  cardinal  number  signified  by  the  numeral  as- 
signed to  the  last  natural  unit  or  component  of  the  col- 
lection in  the  operation  of  counting  it.  Numerals  are 
also  called  numbers.  The  numeral  and  a  word  specify- 
ing the  kind  of  objects  counted  make  what  is  called  a 
concrete  number.  In  distinction  from  this,  a  number  is 
called  an  abstract  number. 

When  children  are  to  count,  the  things  should  be 
sufficiently  distinct  to  be  clearly  and  easily  recognizable  as 
individual,  yet  not  so  disparate  as  to  hinder  the  human 
power  to  make  from  them  an  artificial  individual.  The 
objects  should  not  be  such  as  to  individually  distract  the 
attention  from  the  assemblage  of  them. 

With  little  children  use  a  binary  system.  Build  with 
twos.  Then  go  on,  as  did  the  Romans,  to  a  quinary- 
binary  system,  which  suits  counting  on  the  fingers. 


14          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

In  counting,  an  artificial  individual  may  take  the 
place  of  a  natural  individual.  Children  enjoy  counting 
A  partitioned  Dv  fives.  Inversely,  a  unit  may  be  thought 
unit>  of  as  an  artificial  individual,  composed  of 

subsidiary  individuals,  as  a  dollar  of  100  cents. 

An  interesting  exercise  is  the  instantaneous  recog- 
nition of  the  cardinal,  the  particular  numeric  quality  of 
Number  with-  the  collection,  its  specification  without  count- 
out  counting.  jng  gut  ^jg  power  to  picture  all  the  sep- 
arate individuals  and  to  recognize  the  specific  given  pic- 
ture is  very  limited.  If  it  be  attempted  to  facilitate  this 
recognition  by  arrangement,  the  recognition  may  easily 
become  that  of  form  instead  of  number.  It  is  then  simply 
recognizing  a  shape  which  we  know  should  have  just  so 
many  elements.  Every  teacher  should  remember  when 
using  blocks  in  developing  the  number-concept  that  only 
if  very  few  can  their  number  be  perceived  without  the 
help  of  counting  or  addition.  If  4  blocks  lie  close  their 
number  may  be  perceived  immediately,  but  seven  are 
dealt  with  as  two  groups.  It  is  believed  that  the  limit, 
even  for  adults  and  under  favorable  conditions,  is  about 
4.  We  know  that  even  IIII  was  replaced  by  IV.  Try 
the  children  to  see  if  their  primitive  number  perception, 
that  of  II,  has  grown,  and  how  far. 

In  the  making  of  numeral  words  it  is  necessary  to 
fix  upon  one  after  which  repetition  is  to  begin.  Other- 
Decimal  w*se  tnere  would  be  no  end  to  the  number 

word-  of  different  words  required.    We  have  noted 

numerals.          , ,     ,  ,.  ,  , 

that  the  prehuman  has  narrowed  the  choice, 

by  the  fiveness  of  the  extremities  of  mammalian  limbs, 
to  five,  ten  or  twenty.  The  majority  of  races,  especially 
the  higher,  in  prehistoric  time  chose  ten,  the  number  of 
our  fingers.  Then  was  developed  a  system  to  express 


COUNTING  AND  NUMERALS.  15 

by  a  few  number-names  a  vast  series  of  numbers.  If  we 
interpret  eleven  as  "one  and  ten"  and  twelve  as  "two  and 
ten,"  teen  as  "and  ten,"  ty  as  "tens,"  then  English,  until 
it  took  "million,"  ("great  thousand,"  Latin  mille,  a  thou- 
sand,) bodily  from  the  French  and  Italian,  used  only  a 
dozen  words  in  naming  numbers,  in  making  a  series  of 
word-numerals  with  fixed  order. 

The  systematic  formation  of  numerical  words  is 
called  numeration. 

The  cardinal  number  of  any  finite  set  of  things  is  the 
same  in  whatever  order  we  count  them. 
Invariance  This  is  so   fundamental  a  theorem  of 

of  cardinal,  arithmetic,  it  may  be  well  to  make  its  reali- 
zation more  intuitive. 

That  the  number  of  any  finite  group  of  distinct  things 
is  independent  of  the  order  in  which  they  are  taken,  that 
beginning  with  the  little  finger  of  the  left  hand  and  going 
from  left  to  right,  a  group  of  distinct  things  comes  ulti- 
mately to  the  same  finger  in  whatever  order  they  are 
counted,  follows  simply  from  the  hypothesis  that  they 
are  distinct  things.  If  a  group  of  distinct  things  comes 
to,  say,  five  when  counted  in  a  certain  order,  it  will  come 
to  five  when  counted  in  any  other  order. 

For  a  general  proof  of  this,  take  as  objects  the  letters 
in  the  word  "triangle,"  and  assign  to  each  a  finger,  be- 
ginning with  the  little  finger  of  the  left  hand  and  ending 
with  the  middle  finger  of  the  right  hand.  Each  of  these 
fingers  has  its  own  letter,  and  the  group  of  fingers  thus 
exactly  adequate  is  always  necessary  and  sufficient  for 
counting  this  group  of  letters  in  this  order. 

That  the  same  fingers  are  exactly  adequate  to  touch 
this  same  group  of  letters  in  any  other  order,  say  the 
alphabetical,  follows  because,  being  distinct,  any  pair 


16          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

attached  to  two  of  my  fingers  in  a  certain  order  can  also 
be  attached  to  the  same  two  fingers  in  the  other  order. 

In  the  new  order  I  want  a  to  be  first.  Now  the  letters 
t  and  a  are  by  hypothesis  distinct.  I  can  therefore  inter- 
change the  fingers  to  which  they  are  assigned,  so  that 
each  finger  goes  to  the  object  previously  touched  by  the 
other,  without  using  any  new  fingers  or  setting  free  any 
previously  employed.  The  same  is  true  of  r  and  e,  of 
i  and  g,  etc. 

As  I  go  to  each  one,  I  can  substitute  by  this  process 
the  new  one  which  is  wanted  in  its  stead  in  such  a  way 
that  the  required  new  order  shall  hold  good  behind  me, 
and  since  the  group  is  finite,  I  can  go  on  in  this  way 
until  I  come  to  the  end,  without  changing  the  group  of 
fingers  used  in  counting,  that  is  without  altering  the 
cardinal  number,  in  this  case  8. 

The  group  of  fingers  exactly  adequate  to  touch  a 
group  of  objects  in  any  one  definite  order  is  thus  exactly 
adequate  for  every  order.  But  when  touching  in  one 
definite  order  each  finger  has  its  own  particular  object 
and  each  object  its  own  particular  finger,  so  that  the 
group  of  fingers  exactly  adequate  for  one  peculiar  order 
is  always  necessary  and  sufficient  for  that  one  order. 
But  we  have  shown  it  then  exactly  adequate  for  every 
order;  therefore  it  is  necessary  and  sufficient  for  every 
order. 


CHAPTER  IV. 

GENESIS  OF  OUR  NUMBER  NOTATION. 

The  systematic  decimal  system  in  accordance  with 
which,  even  in  the  times  of  our  prehistoric  ancestors,  a 
Positional  f£w  number  names  were  used  to  build  all 
counting.  numeral  words,  is  paralleled  by  the  proce- 
dure, even  at  the  present  day,  of  those  Africans  who  in 
counting  use  a  row  of  men  as  follows rjihe  first  begins 
with  the  little  finger  of  the  left  hand,  and  indicates,  by 
raising  it  and  pointing  or  touching,  the  assignment  of 
this  finger  as  representative  of  a  certain  individual  from 
the  group  to  be  counted;  his  next  finger  he  assigns  to 
another  individual;  and  so  on  until  all  his  fingers  are 
raised.  And  now  the  second  man  raises  the  little  finger 
of  his  left  hand  as  representative  of  this  whole  ten,  and 
the  first  man,  thus  relieved,  closes  his  fingers  and  begins 
over  again.  When  this  has  been  repeated  ten  times,  the 
second  man  has  all  his  fingers  up,  and  is  then  relieved 
by  one  finger  of  the  third  man,  which  finger  therefore 
represents  a  hundred ;  and  so  on  to  a  finger  of  the  fourth 
man,  which  represents  a  thousand,  and  to  a  finger  of  the 
fifth  man,  which  represents  a  myriad  (ten  thousand). 

An  advance  on  this  actual  use  of  fingers  with  a  posi- 
tional value  depending  only  on  the  man's  place  in  the 
row,  is  seen  in  the  widely  occurring  abacus, 
a  rough  instance  of  which  is  just  a  row  of 
grooves  in  which  pebbles  can  slide.  With  most  races,  as 


18          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

with  the  Egyptians,  Greeks,  Japanese,  the  grooves  or  col- 
umns are  vertical,  like  a  row  of  men.  The  counters  in  the 
right-most  column  correspond  to  the  fingers  of  the  man 
who  actually  touches  or  checks  off  the  individuals 
counted;  it  is  the  units  column. 

But  in  the  abacus  a  simplification  occurs.  One  finger 
of  the  second  man  is  raised  to  picture  the  whole  ten 
fingers  of  the  first  man,  so  that  he  may  lower  them  and 
begin  again  to  use  them  in  representing  individuals.  Thus 
there  are  two  designations  for  ten,  either  all  the  fingers 
of  the  first  man  or  one  finger  of  the  second  man.  The 
abacus  omits  the  first  of  these  equivalents,  and  so  each 
column  contains  only  nine  counters. 

For  purposes  of  counting,  a  group  of  objects  can  be 
represented  by  a  graphic  picture  so  simple  that  it  can  be 
Recorded  produced  whenever  wanted  by  just  making 
symbols.  a  mark  for  Gacfo  distinct  object.  Thus  the 
marks  I,  II,  III,  IIII,  picture  the  simplest  groups  with  a 
permanence  beyond  gesture  or  word;  and  for  many  im- 
portant purposes,  one  of  these  stroke-diagrams,  though 
composed  of  individuals  all  alike,  is  an  absolutely  per- 
fect picture,  as  accurate  as  the  latest  photograph,  of  any 
group  of  real  things  no  matter  how  unlike. 

The  ancient  Egyptians  denoted  all  numbers  under  ten 
by  the  corresponding  number  of  strokes;  but  with  ten 
a  new  symbol  was  introduced.  The  Romans  regularly 
used  strokes  for  numbers  under  five,  using  V  for  five. 
The  ancient  Greeks  and  Romans  both  however  indicated 
numbers  by  simple  strokes  as  high  as  ten.  The  Aztecs 
carried  this  system  as  high  as  twenty,  but  they  used  a 
small  circle  in  place  of  the  straight  stroke.  I  have  seen 
the  same  thing  done  in  Japan. 

Each  stroke  of  such  a  picture-group  may  be  called  a 


GENESIS  OF  OUR  NUMBER  NOTATION.  19 

unit.     Each  group  of  such  units  will  correspond  always 
to  the  same  group  of  fingers,  to  the  same  numeral  word. 

Though  to  this  primitive  graphic  system  of  number- 
pictures  there  is  no  limit,  yet  it  soon  becomes  cumbrous. 
The  Hindu  Abbreviations  naturally  arise.  Those  the 
numerals.  world  now  uses,  the  Hindu  numerals,  have 
been  traced  back  to  inscriptions  in  India  probably  dating 
from  the  early  part  of  the  second  century  B.  C. 

The  oldest  inscription  using  them  positionally  with 
local  value  and  developed  form  is  of  595  A.  D.  The 
Egyptians  had  no  positional  notation  for  number,  though 
they  had  a  hieroglyph  for  nothing,  which  they  substituted 
for  one  side  when  applying  their  formula  for  a  quadri- 
lateral to  a  triangle.  The  Babylonians  had  a  sign  of 
this  kind,  not  used  in  calculation,  consisting  of  two  angu- 
lar marks,  one  above  the  other.  About  A.  D.  130,  Ptol- 
emy in  Alexandria  used,  in  his  Almagest,  the  Babylonian 
sexagesimal  fractions,  and  designated  voids  by  the  first 
letter  of  the  word  ovSe'v,  nothing.  This  letter  was  not 
used  as  a  zero. 

M.  F.  Nau  gives  in  French  translation  in  Journal 
asiatique,  Vol.  16  (10th  series),  1910,  pp.  225-227,  a 
quotation  from  Severus  Sebokt,  of  Quennesra,  on  the 
Euphrates,  near  Diarbekr,  written  in  662  A.  D.,  more 
than  two  centuries  before  the  earliest  known  appearance 
of  the  numerals  in  Europe: 

"I  refrain  from  speaking  of  the  science  of  the  Hin- 
dus, who  are  not  Syrians,  of  their  subtile  discoveries  in 
this  science  of  astronomy — more  ingenious  than  those 
of  the  Greeks  and  even  of  the  Babylonians — and  of  their 
facile  method  of  calculating  and  computing,  which  sur- 
passes words.  I  mean  that  made  with  nine  symbols." 


20  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

But  probably  a  long  time  was  yet  to  pass  before  the 
creation  of  the  most  useful  symbol  in  the  world,  the 
naught,  the  zero,  not  merely  a  sign  for  noth- 
ing, but  a  mark  for  the  absence  of  quantity, 
the  cipher,  whose  first  known  use  in  ring  form  in  a 
document  is  in  738  A.  D.* 

This  little  ellipse,  picture  for  airy  nothing,  is  an  indis- 
pensable corner-stone  of  modern  civilization.  It  is  an 
Ariel  lending  magic  powers  of  computation,  promoting 
our  kindergarten  babies  at  once  to  an  equality  with  Cae- 
sar, Plato  or  Paul  in  matters  arithmetical. 

The  user  of  an  abacus  might  instead  rule  columns  on 
paper  and  write  in  them  the  number  of  pebbles  or  coun- 
ters. But  zero,  0,  shows  an  empty  column  and  so  at 
once  relieves  us  of  the  need  of  ruling  the  columns,  or 
using  the  abacus.  Modern  arithmetic  comes  from  ancient 
counting  on  the  columns  of  the  abacus,  immeasurably 
improved  by  the  creation  of  a  symbol  for  an  empty  col- 
umn. 

The  importance  of  the  creation  of  the  zero  mark  can 
never  be  exaggerated.  This  giving  to  airy  nothing  not 
merely  a  local  habitation  and  a  name,  a  picture,  a  symbol, 
but  helpful  power,  is  characteristic  of  the  Hindu  race 
whence  it  sprang.  It  is  like  coining  the  Nirvana  into 
dynamos.  No  single  mathematical  creation  has  been  more 
potent  for  the  general  on-go  of  intelligence  and  power. 
From  the  second  half  of  the  eighth  century  Hindu  writ- 
ings were  current  at  Bagdad.  After  that  the  Arabs  knew 
positional  notation.  They  called  the  zero  gifr.  The  Arab 
word,  a  substantive  use  of  the  adjective  gifr  ("empty"), 
was  simply  a  translation  of  the  Sanskrit  name  sunya, 

*  E.  C.  Bayley,  1882  Doubted  by  G.  F.  Hill,  1910,  who  substi- 
tuted an  inscription  of  876  A.  D. 


GENESIS  OF  OUR  NUMBER  NOTATION.  21 

literally  "empty."  It  gave  birth  to  the  low-Latin  zephi- 
rum  or  zefirum  (used  by  Leonard  of  Pisa,  1202),  whence 
the  Italian  form  zefiro,  contracted  to  zefro,  and  (1307) 
zeuero,  then  zero,  whose  introduction  in  print  goes  back 
to  the  15th  century  (1491). 

In  the  oldest  known  French  treatise  on  algorithm 
(author  unknown,  of  the  thirteenth  century)  we  read, 
"iusca  le  darraine  ki  est  appellee  cifre  0."  In  the  thir- 
teenth century  in  Latin  the  word  cifra  for  "naught"  is 
met  in  Jordan  Nemorarius  and  in  Sacrabosco  who  wrote 
at  Paris  about  1240. 

In  MS.  Egerton  2622,  one  of  the  earliest  arithmetics 
in  our  language,  on  leaf  \2>7b,  we  read: 

"Nil  cifra  significat  sed  dat  signare  sequent!. 

"Expone  this  verse.  A  cifre  tokens  noyt,  bot  he 
makes  the  figure  to  betoken  that  comes  aftur  hym  more 
than  he  schuld  &  he  were  away,  as  thus  10.  here  the 
figure  of  one  tokens  ten.  it  may  happe  aftur  a  cifre 
schuld  come  a  nothur  cifre,  as  thus  200." 

Maximus  Planudes  (1330)  uses  tziphra.  Euler  used 
(1783)  in  Latin  the  word  cyphra.  We  still  say  "cipher" 
or  "cypher."  In  German  Ziffer  has  taken  a  more  gen- 
eral meaning,  as  has  the  equivalent  French  word  chiffre, 
the  most  important  numeral  coming  to  mean  any.  The 
oldest  coin  positionally  dated  is  of  1458. 

Zero,  originally  the  sign  of  a  blank  or  nil  or  vacant 
column,  may  be  looked  upon  as  indicating  that  a  class  is 
void,  containing  no  object  whatever,  that  it  is  the  null 
class.  Thus  it  is  one  of  the  answers  to  the  question, 
"How  many?",  and  so  is  a  cardinal.  It  is  also  given  a 
place  in  the  ordinal  series  of  natural  numbers,  and  is 
chief  in  the  series  of  algebraic  numbers.  Only  in  the 


22          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

sixteenth  century  does  naught  appear  as  common  sym- 
bol for  all  differences  in  which  minuend  and  subtrahend 
are  equal,  and  thus  show  itself  as  ready  for  its  second 
great  application,  to  standardize  algebraic  forms. 

By  the  first  meaning  of  cipher,  "empty,"  we  have 
20  =  twain  ten,  but  2  +  0  =  2.  Hankel,  1867,  calls  modu- 
lus of  an  operation  that  which  combined  by  the  operation 
with  something  leaves  this  unchanged.  So  to-day  we 
use  nine  digits  and  have  no  digit  corresponding  to  the 
Roman  X,  for  X  is  all  the  fingers  of  the  first  man,  while 
we,  like  the  abacus,  use  10,  which  is  one  finger  of  the 
second  man.  Thus  the  ten,  hundred,  thousand  are  only 
expressed  by  the  position  of  the  number  which  multiplies 
them. 

In  the  written  numeral  IIII,  we  still  see  in  the  symbol 
the  units  of  which  the  fourfold  unit  four  is  composed. 
Later  abbreviation  veils  the  constituent  units,  but  their 
independence  and  all-alike-ness  remain  fundamental,  giv- 
ing to  cardinal  number  its  independence  of  the  order  in 
which  the  things  are  enumerated. 

The  use  of  the  digits  (Latin,  digitus,  a  "finger"),  the 
substitution  of  a  single  symbol  for  each  of  the  first  nine 
Our  present  picture-groups,  and  that  splendid  creation  of 
notation.  the  Hindus,  the  zero,  0,  naught,  cipher, 
made  possible  our  present  notation  for  number.  This 
still  has  a  base,  ten,  in  which  the  sins  of  our  fathers,  the 
mammals,  are  visited  on  their  children.  Its  perfection 
is  in  its  use  of  position  with  digits  and  zero,  a  positional 
notation  for  number,  which  the  decimal  point  (or  unital 
point)  empowers  to  run  down  below  the  units,  giving  the 
indispensable  decimals. 

This  positional  notation  for  number  consists  in  the 
very  refined  artifice  of  representing  every  number  as  a 


GENESIS  OF  OUR  NUMBER  NOTATION.  23 

sum  of  terms  expressed  by  a  row  of  digits  each  standing 
for  a  product  of  two  factors,  one  factor  the  intrinsic, 
the  face  factor,  indicated  by  the  digit  itself,  the  other 
factor,  the  local,  the  place  factor,  indicated  by  the  place 
of  this  digit  in  the  row,  the  local  factor  being  a  power 
of  the  base,  for  units'  place,  or  column  b°  or  one,  for 
the  next  place  to  the  left  b1  or  b  (the  base),  to  the  right 
b~l  or  \/b,  etc.  The  summation  of  these  binary  products 
is  indicated  by  the  juxtaposition  in  the  row  of  the  digits 
representing  them  by  their  form  and  their  place  in  the 
row  with  reference  to  units'  place. 

Calculus,  (Latin,  "a  pebble"),  ciphering,  which  thus 
by  the  aid  of  zero  attains  an  ease  and  facility  which 
would  have  astounded  the  antique  world,  consists  in  com- 
bining given  numbers  according  to  fixed  laws  to  find 
certain  resulting  numbers. 

Teaching  is  to  enable  the  ordinary  child  to  do  what 
the  genius  has  done  untaught. 

A  Hindu  genius  created  the  zero.  The  common,  even 
the  stupid,  child  is  now  to  be  taught  to  understand  and 
use  this  wonderful  creation  just  as  it  is  taught  to  use 
the  telephone.  So  the  teacher  incites,  provokes  the  self- 
activity  of  the  child's  mind  and  guides  it  and  confirms 
it,  stopping  this  kaleidoscope  at  a  certain  turn,  when  the 
evershifting  picture  is  near  enough  for  life  to  the  picture 
in  the  teacher's  mind. 

Without  theory,  no  practice,  yet  need  not  the  theory 
be  conscious.  There  is  a  logic  of  it,  yet  the  child  need 
not  necessarily  know,  had  perhaps  better  not  know,  that 
logic.  The  teacher  should  know,  the  child  practise. 

It  is  striking  to  realize  the  centuries  that  passed  after 
the  present  system  of  number-naming,  numeration,  had 


24          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

been  developed,  before  it  had  analogous,  adequate  sym- 
bolization,  adequate  written  notation. 

As  compared  with  their  number-names,  how  bungling 
the  Greek  and  Roman  numerals,  how  arithmetically  help- 
less the  men  of  classic  antiquity  for  lack  of  just  one  writ- 
ten symbol,  the  Hindu  naught,  giving  us  a  written  system 
which,  except  for  its  base  ten,  seems  to  be  final  and  for 
all  time,  a  world  sign-language  more  perspicuous  and 
compendious  than  any  word-language.  That  prehuman 
parasite,  the  ten,  is  fixed  on  us  like  an  Old  Man  of  the 
Sea,  else  we  could  take  the  easily  superior  base  twelve. 
The  number  of  digit  figures  required  is  one  less  than  the 
base;  since  10  represents  the  base,  whatever  it  be. 

In  each  case  the  prebasal  figures,  by  help  of  the  zero, 
always  express  as  written  in  succession  to  left  or  to  right 
of  the  units  place  (fixed  by  the  unital  point)  multiples 
of  ascending  and  descending  powers  of  the  base.  But 
while  the  two  and  six  of  twelve  are  like  the  two  and  five 
of  ten,  yet  twelve  has  three  and  four  besides  as  divisors, 
as  submultiples,  for  which  tremendous  advantage  ten 
offers  no  equivalent  whatsoever.  The  prehuman  imposi- 
tion of  ten  as  base,  disbarring  twelve,  is  thus  a  permanent 
clog  on  human  arithmetic. 

The  mere  numerals,  1,  2,  3,....  or  the  numeral 
words,  "one,"  "two,"  "three," ....  are  signs  for  what 
are  called  "natural  numbers,"  or  positive  integers.  In- 
teger with  us  shall  always  mean  positive  integer.  If 
pure  numbers,  integers,  have  an  intrinsic  order,  so  do 
these,  their  symbols. 

The  unending  series,  1,  2,  3,  4,  5,. ...  or  one,  two, 
three,  four,  five,..  ..is  called  the  "natural  scale,"  or 
the  scale  of  the  natural  numbers,  or  the  number  series. 
Each  symbol  in  it,  besides  its  ordinal,  positional  sig- 


GENESIS  OF  OUR  NUMBER  NOTATION.  25 

nificance  in  the  sequence  of  symbols,  is  used  also  to  in- 
dicate the  cardinal  number  of  the  symbols  in  the  piece 
of  the  scale  it  ends,  and  so  of  any  group  correlated  to 
that  piece. 

Thus  the  ordinal  system  is  the  original  from  which 
the  cardinal  system  is  derived. 

In  the  primary  ordinal  system  the  symbols  refer  to 
the  individual  objects,  while  in  the  derived  cardinal  sys- 
tem these  same  symbols  refer  to  the  successively  larger 
sets  whose  names  are  determined  as  the  name  of  the 
last  individual  counted  ordinally. 


CHAPTER  V. 

THE  TWO  DIRECT  OPERATIONS,  ADDITION 
AND  MULTIPLICATION. 

The  symbolic  representation  of  numbers  and  ways 

of  combining  numbers  comes  under  the  head 
Notation.  .    "  . 

of  what  is  called  notation. 

The  natural  numbers,  as  shown  in  the  primitive  nu- 
meral pictures,  I,  II,  III,  IIII,  begin  with  a  single  unit, 
and,  cardinally  considered,  are  changed  to  the  next  al- 
ways by  taking  another  single  unit. 

A  number,  an  integer,  is  said  to  be  equal  to,  or  the 
same  as,  a  number  otherwise  expressed,  when  their  units 
The  symbol  being  counted  come  to  the  same  finger,  the 
—•  same  numeral  word.  The  symbol  =,  read 

equals,  is  called  the  sign  of  equality,  and  takes  the  part 
of  verb  in  this  symbolic  language.  It  was  invented  by 
an  Englishman,  Robert  Recorde,  replacing  in  his  algebra, 
The  Whetstone  of  Witte*  the  sign  z  used  for  equality 
in  his  arithmetic,  The  Grounde  of  Artes,  1540.  Equality 
is  a  relation  reflexive,  symmetric,  invertible.  Equality 
is  a  mutual  relation  of  its  two  members.  If  x=y,  then 
y-x.  Equality  is  a  transitive  relation.  If  x-y  and 
y—z,  then  x—z.  A  symbolic  sentence  using  this  verb 
is  called  an  equality. 

Ordinally,  x=y  means  that  x  and  y  denote  the  same 

*  London  (no  date,  preface  1557). 


ADDITION  AND  MULTIPLICATION.  27 

number  in  the  natural  scale.  Formally,  x-y  means  that 
either  can  at  will  be  substituted  for  the  other  anywhere. 

When  the  process  of  counting  the  units  of  one  num- 
ber simultaneously  one-to-one  with  units  of  a  second 
number  ends  because  no  unit  of  the  second 
number  remains  uncounted,  but  the  units 
of  the  first  number  are  not  all  counted,  then  the  first 
number  is  said  to  contain  more  units  than  the  second 
number,  and  the  second  number  is  said  to  contain  less 
units  than  the  first. 

If  a  number  contains  more  units  than  a  second,  it  is 
called  greater  than  this  second,  which  is  called  the  lesser. 
By  successively  incorporating  single  units  with  the  lesser 
of  two  primitive  numbers  we  can  make  the  greater. 

Thomas  Harriot*  (1560-1621),  tutor  to  Sir  Walter 
Raleigh  and  one  of  "the  three  magi  of  the  Earl  of  North- 
umberland," devised  the  symbol  >,  published  1631,  read 
"is  greater  than,"  and  called  the  sign  of  inequality.  In- 
equality is  a  sensed  relation.  Turned  thus  <  its  symbol 
is  read  "is  less  than."  Inequality  in  the  same  sense  is 
transitive.  If  x  >  y  and  y  >  2,  then  x  >  z. 

Since  the  result  of  counting  is  independent  of  the 
order  of  the  individuals  counted,  therefore  of  two  un- 
equal natural  numbers  the  one  once  found  greater  is 
always  the  greater.  Without  knowing  the  number  n, 
we  can  write  "either  n>5,  or  n=5,  or  n  <  5."  Any 
number  which  succeeds  another  in  the  natural  scale  is 
greater  than  this  other.  Ordinally,  x  <  y  means  that  x 
precedes  y  in  the  scale. 

*  Harriot  was  sent  to  America  by  Raleigh  in  the  year  1585. 
He  made  the  first  survey  of  Virginia  and  North  Carolina,  the  maps 
of  these  being  subsequently  presented  to  Queen  Elizabeth.  He  started 
the  standardizing  of  algebraic  forms  and  the  theory  of  functions  by 
writing  every  equation  as  a  function  equal  to  zero. 


28  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

When  by  any  definite  process  we  select  one  or  more 
elements  of  any  aggregate  A,  these  form  another  aggre- 
gate B,  called  a  part  of  A.  If  any  element  of  A  remains 
unselected,  B  is  called  a  proper  part  of  A.  It  is  possible 
for  an  aggregate  to  be  equivalent  to  a  proper  part  of  it- 
self;  the  aggregate  is  then  called  infinite.  For  example: 
for  every  number  there  is  an  even  number;  again,  for 
every  point  on  a  foot  there  is  a  point  on  an  inch. 

When  we  can  get  a  third  number  from  two  given 
numbers  by  a  definite  operation,  the  two  given  numbers 

joined  by  the  sign  for  the  operation  and 
Parentheses.    J      ,          '. 

enclosed   m  parentheses  may  be  taken  to 

mean  the  result  of  that  combination.  The  result  can  now 
be  again  combined  with  another  given  number,  and  so 
we  may  get  combinations  of  several  numbers  though 
each  operation  is  performed  only  with  two. 

Parentheses  indicate  that  neither  of  the  two  numbers 
enclosed,  but  only  the  number  produced  by  their  combina- 
tion, is  related  to  anything  outside  the  parentheses. 

Parentheses  (first  used  by  the  Flemish  geometer  Al- 
bert Girard  in  1629)  may  without  ambiguity  be  omitted: 

First,  When  of  two  operations  of  like  rank  the  pre- 
ceding (going  from  left  to  right)  is  to  be  first  carried 
out; 

Second,  When  of  two  operations  of  unlike  rank  the 
higher  is  the  first  to  be  carried  out. 

The  representation  of  one  number  by  others  with 
symbols  of  combination  and  operation  is  called  an  ex- 
pression.    By  enclosing  it  in  parentheses, 
Expressions.    r  .  J 

any  expression  however  complex  in  any  way 

representing  a  number,  may  be  operated  upon  as  if  it 
were  a  single  symbol  of  that  number.  If  an  expression 
already  involving  parentheses  is  enclosed  in  parentheses, 


ADDITION  AND  MULTIPLICATION.  29 

each  pair,  to  distinguish  it,  can  be  made  different  in  siae 
or  shape.  The  three  most  usual  forms  are  the  parenthesis 
(,  the  bracket  [,  and  the  brace  {.  In  translating  the  ex- 
pression into  English,  (  should  be  called  first  parenthesis, 
and  )  second  parenthesis ;  [  first  bracket,  ]  second  bracket ; 
{  first  brace,  }  second  brace. 

No  change  of  resulting  value  is  made  in  any  expres- 
sion by  substituting  for  any  number  its  equal  however 

expressed.     From  this  it  follows  that  two 
Substitution.  ...  t 

numbers  each  equal  to  a  third  are  equal  to 

one  another.  This  process,  putting  one  expression  for 
another,  substitution,  is  a  primitive  yet  most  important 
proceeding.  A  single  symbol  may  be  substituted  for  any 
expression  whatever. 

Permutation  consists  in  a  simultaneous  carrying  out 
of  mutual  substitution,  interchange.  Thus  a  and  b  in  an 
expression,  as  abc,  are  permuted  when  they  are  inter- 
changed, giving  bac.  More  than  two  symbols  are  per- 
muted when  each  is  replaced  by  one  of  the  others,  as 
in  abc  giving  bca  or  cab. 

Suppose  we  have  two  natural  numbers  written  in 
their  primitive  form,  as  III  and  IIII;  if  we  write  all 

these  units  in  one  row  we  indicate  another 
Addition. 

natural  number;  and  the  process  of  getting 

from  two  numbers  the  number  belonging  to  the  group 
formed  by  putting  together  their  groups  to  make  a  single 
group  is  called  addition.  This  operation  of  incorporating 
other  units  into  the  preceding  diagram  is  indicated  by 
a  symbol  first  met  in  print  in  the  arithmetic  by  John 
Widman,  (Leipsic,  1489),  a  little  Greek  cross,  +,  read 
plus. 

If  one  artificial  individual  be  combined  with  another 
to  give  a  new  artificial  individual  in  which  each  unit  of 


30          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

the  components  appears  retaining  its  natural  indepen- 
dence and  natural  individuality,  while  the  artificial  indi- 
viduality of  the  two  components  vanishes,  the  number 
of  the  new  artificial  individual  is  called  the  sum  of  the 
numbers  of  the  two  components,  and  is  said  to  be  ob- 
tained by  adding  these  two  numbers  (the  terms  or  sum- 
mands).  The  first  of  two  summands  may  be  called  the 
augment;  the  second,  the  increment.  The  sum  of  two 
numbers,  two  terms,  is  the  numeric  attribute  of  the  total 
system  constituted  of  two  partial  systems  to  which  the 
two  terms  respectively  pertain. 

In  the  child  as  in  the  savage,  the  number  idea  is  not 
dissociated  from  the  group  it  characterizes.  But  educa- 
tion should  help  on  the  stage  where  the  number  exists 
as  an  independent  concept,  say  the  number  five  with  its 
own  characteristics,  its  own  life.  Therefore  we  have 
number-science,  pure  arithmetic.  So  though  it  might  per- 
haps be  argued  that  there  is  only  one  number  5,  yet  we 
may  properly  speak  of  combining  5  with  5  so  as  to  retain 
the  units  unaffected  while  the  fiveness  vanishes  in  the 
compound,  the  sum,  10. 

Addition  is  a  taking  together  of  the  units  of  two  num- 
bers to  constitute  the  units  of  a  third,  their  sum.  This 
may  be  obtained  by  a  repetition  of  the  operation  of  form- 
ing a  new  number  from  an  old  by  taking  with  it  one 
more  unit;  thus  3  +  2  =  3  +  1  +  1. 

If  given  numbers  are  written  as  groups  of  units,  e.  g. 
(exempli  gratia) ,  2=1  +  1,  3  =  1  +  1  +  1,  the  result  of 
adding  is  obtained  by  writing  together  these  rows  of 
units,  e.g.,  2  +  3=(l +  !)  +  (! +  1  +  1)  =  1  +  1 +  1  +  1 +1=5. 

Since  cardinal  number  is  independent  of  the  order 
of  counting,  therefore  in  any  natural  number  expressed 


ADDITION  AND  MULTIPLICATION.  31 

in  its  primitive  form,  as  IIII,  the  permutation  of  any 
pair  of  units  produces  neither  apparent  nor  real  change. 

The  units  of  numeration  are  completely  interchange- 
able. Therefore  we  may  say  adding  numbers  is  finding 
one  number  which  contains  in  itself  as  many  units  as  the 
given  numbers  taken  together. 

In  defining  addition,  we  need  make  no  mention  of  the 
order  in  which  the  given  numbers  are  taken  to  make  the 
sum.  A  sum  is  independent  of  the  order  of  its  parts 
or  terms.  This  is  an  immediate  consequence  of  the  theo- 
rem of  the  invariance  of  the  number  of  a  set.  For  a 
change  in  the  order  of  the  parts  added  is  only  a  change 
in  the  order  of  the  units,  which  change  is  without  in- 
fluence when  all  are  counted  together. 

To  write  in  symbols,  in  the  universal  language  of 
mathematics,  that  addition  is  an  operation  unaffected  by 
permutation  of  the  order  of  the  parts  added,  though 
applied  to  any  numbers  whatsoever,  we  cannot  use  nu- 
merals, since  numerals  are  always  absolutely  definite, 
particular.  If,  following  Vieta's  book  of  1591,  we  use 
letters  as  general  symbols  to  denote  numbers  left  other- 
wise indefinite,  we  may  write  a  to  represent  the  first 
number  not  only  in  the  sum  2  +  3,  but  in  the  sum  4  +  1 
and  in  the  sum  of  any  two  numbers.  Taking  b  for  a 
second  number,  the  symbolic  sentence  a  +  b  =  b  +  a  is  a 
statement  about  all  numbers  whatsoever.  It  says,  addi- 
tion is  a  commutative  operation. 

The  words  commutative  and  distributive  were  used 
for  the  first  time  by  F.  J.  Servois  in  1813. 

The  previous  grouping  of  the  parts  added  has  no 
effect  upon  the  sum.  Brackets  occurring  in  an  indicated 
sum  may  be  omitted  as  not  affecting  the  result.  The 
general  statement  or  formula  (a  +  b)  +c  =  a+  (b  +  c)  says, 


32  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

addition  is  an  associative  operation,  an  operation  having 
associative  freedom. 

Rowan  Hamilton  in  1844  first  explicitly  stated  and 
named  the  associative  law.  For  addition  it  follows  from 
the  theorem  of  the  invariance  of  the  number  of  a  group. 

Equalities  having  to  do  only  with  the  very  nature 

of  the  operations  involved,  and  not  at  all 

Formulas.  .  ,     .  .     ,  „    , 

with  the  particular  numbers  used  are  called 

formulas. 

A  formula  is  characterized  by  the  fact  that  for  any 
letter  in  it  any  number  whatsoever  may  be  substituted 
without  destroying  the  equality  or  restricting  the  values 
of  any  other  letter.  In  a  formula  a  letter  as  symbol  for 
any  number  may  be  replaced  not  only  by  any  digital  num- 
ber, but  also  by  any  other  symbol  for  a  number  whether 
simple  or  compound,  in  the  last  case  bracketed.  Thus 
a  +  b  =  b  +  a  gives  (a+c)  +b  =  b+  (a+c).  So  from  a 
formula  we  can  get  an  indefinite  number  of  formulas 
and  special  numerical  equations. 

Each  side  or  member  of  a  formula  expresses  a  method 
of  reckoning  a  number,  and  the  formula  says  that  both 
reckonings  produce  the  same  result.  A  formula  trans- 
lated from  symbols  into  words  gives  a  rule.  As  equality 
is  a  mutual  relation  always  invertible,  a  formula  will 
usually  give  two  rules,  since  its  second  member  may  be 
read  first. 

By  definition,  from  the  inequality  a  >  b  we  know 
that  a  could  be  obtained  by  adding  units  to  b.  Calling 
this  unknown  group  of  units  n,  we  have  a  =  b  +  n. 

Inversely,  if  a=b  +  n  then  a  >  b,  that  is  a  sum  of  finite 
natural  numbers  is  always  greater  than  one  of  its  parts. 
A  sum  increases  if  either  of  its  parts  increases. 


ADDITION  AND  MULTIPLICATION.  33 

Addition  may  also  be  defined  and  its  properties  es- 
Ordinal  tablished  from  the  ordinal  view-point. 

addition.  start  from  the  natural  scale.     To  add 

1  to  the  number  x  is  to  replace  x  by  the  next  following 
ordinal.     So  if  we  know  x,  we  know  x+\. 

When  we  have  defined  adding  some  particular  num- 
ber a  to  x,  when  we  have  defined  the  operation  x  +  a,  the 
operation  x+(a+\}  shall  be  defined  by  the  formula 

(1)  .....  .r+(a+  l)  =  (*  +  a)  +  l. 

We  shall  know  then  what  ;r+(a  +  l)  is  when  we 
know  what  x  +  a  is,  and  as  we  have,  to  start  with,  defined 
what  x+\  is,  we  thus  have  successively  and  "by  recur- 
rence" the  operations  x  +  2,  x  +  3,  etc. 

The  sum  a  +  b  is  thus  defined  ordinally  as  the  bth  term 
after  the  ath. 

It  serves  to  represent  conventionally  a  new  number 
univocally  deduced  by  a  definite  given  procedure  from 
the  numbers  summed  or  added  together. 

Associativity  :  a+  (b  +  c)  =  (a  +  b)  +c. 

This  theorem  is  by  definition  true  for  c=l,  since,  by 
Properties  formula  (1),  a+  (b  +  1)  =  (a  +  b)  +  1.  Now 
of  addition,  supposing  the  theorem  true  for  c  =  y,  it  will 
be  true  for  c  =  y  +  l.  For  supposing 


it  follows  that 

(2)  .  .  .  .  [  (fl  +  fe)  +  y]  +  1  = 

which  is  only  adding  one  to  the  same  number,  to  equal 
numbers. 

Now  by  definition  (  1  )  the  first  member  of  this  equa- 
tion (2) 

....  (3), 


34  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

as  we  recognize  that  it  should  be,  since  y  is  the  number 
preceding  y+l. 

But  by  the  same  formula   (1),  read  backward,  the 
second  member  of  equation  (2) 


as  we  see  it  should  be,  since  b  +  y  is  the  number  preceding 
b  +  y  +  l.    But  again  by  (1),  the  second  member  of  (4), 


Therefore  [by  (5),  (4)  and  (3)],  (2)  may  be  writ- 
ten, 


Hence  the  theorem  is  true  for  c  =  y+l. 

Being  true  for  c  =  l,  we  thus  see  successively  that 
so  it  is  for  c  =  2,  for  c  =  3,  etc. 

This  is  a  proof  by  mathematical  induction  or  demon- 
stration by  recurrence,  a  procedure  first  explicitly  used, 
although  without  a  general  enunciation,  by  Maurolycus 
in  his  work,  Arithmetic  orum  libri  duo  (Venice,  1575). 

Commutativity  :  1°....  a  +  l  =  l+a. 
This  theorem  is  identically  true  for  a=l. 
Now  we  can  verify  that  if  it  is  true  for  a  —  y  it  will 
be  true  for  a  =  y  +  l  ;  for  then 


by  associativity.    But  it  is  true  for  a  =  1,  therefore  it  will 
be  true  for  a  =  2,  for  a  =  3,  etc. 


2°  ---- 

This  has  just  been  demonstrated  for  &  =  1;  it  can  be 
verified  that  if  it  is  true  for  b-x,  it  will  be  true  for 
For,  if  true  for  b  =  x,  then  we  have  by  hypoth- 


ADDITION  AND  MULTIPLICATION.  35 


esis  a  +  x  =  x  +  a;  whence,  by  formula  (1),  by  1°  and  asso- 
ciativity, o+  (-*•+!)  =  (a  +  .*•)  +  !  =  (.*•  +  a)  +  l=#+(a+l) 
=*+(!  +  a)  »(*+!)+  a. 

The  proposition  is  therefore  established  by  recurrence. 

Sums  in  which  all  the  parts  are  equal  frequently  occur. 
Such  additions  are  often  laborious  and  liable  to  error. 
Multiplica-  But  such  a  sum  is  determined  if  we  know 
tion.  one  0£  the  equai  parts  and  the  number  of 

parts.  The  operation  of  combining  these  two  numbers  to 
get  the  result  is  called  multiplication;  the  result  is  then 
called  the  product.  The  part  repeated  is  called  the  multipli- 
cand, and  the  number  which  indicates  how  often  it  occurs 
is  called  the  multiplier.  Multiplicand  and  multiplier  are 
each  factors  of  the  product.  Such  a  product  is  a  multiple 
of  each  of  its  factors.  In  forming  such  a  product,  the 
multiplicand  is  taken  once  as  summand  for  each  unit  in 
the  multiplier.  More  generally,  a  product  is  the  number 
related  to  the  multiplicand  as  the  multiplier  to  the  unit. 

Following  Wm.  Oughtred  (1631),  we  use  the  sign 
x  to  denote  multiplication,  writing  it  before  the  multiplier 
but  after  the  multiplicand.  Thus  1  xlO,  read  one  multi- 
plied by  ten,  or  simply  one  by  ten,  stands  for  the  product 
of  the  multiplication  of  1  by  10,  which  by  definition 
equals  10.  The  multiplication  sign  may  be  omitted  when 
the  product  cannot  reasonably  be  confounded  with  any- 
thing else,  thus  la  means  1  xa,  read  one  by  a,  which  by 
definition  equals  a. 

From  our  definition  also  axl,  that  is  a  multiplied 
by  1,  must  equal  a. 

Commutativity.  Multiplication  of  a  number  by  a  num- 
ber is  commutative. 

Multiplier  and  multiplicand  may  be  interchanged  with- 
out altering  the  product. 


36  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

11111  For  if  we  have  a  rectangular  array  of 

11111  a  rows  each  containing  b  units,  it  is  also  b 
11111  columns  each  containing  a  units. 

Therefore  bxa  =  axb. 

Taking  apposition  to  mean  successive  multiplication, 
for  example,  abode  =  {[(ab}c]d}e,  calling  the  numbers 
involved  factors,  and  the  result  their  product,  we  may 
prove  that  commutative  freedom  extends  to  any  or  all 
factors  in  any  product. 

For  changing  the  order  of  a  pair  of  factors  which 
are  next  one  another  does  not  alter  the  product.  abcd- 
acbd. 

For  c  rows  of  a's,  each  row  containing 

a  a  a  a  a     b  of  them,  is  b  columns  of  a's,  each  con- 

a  a  a  a  a     taining  c  of  them.     So  c  groups  of  ab  units 

a  a  a  a  a     comes  to  the  same  number  as  b  groups  of  ac 

units. 

This  reasoning  holds  no  matter  how  many  factors 
come  before  or  after  the  interchanged  pair.  For  example 

abcdefg=abc  ed  fg, 

since  in  this  case  the  product  abc  simply  takes  the  place 
which  the  number  a  had  before.  And  e  rows  with  d 
times  abc  in  each  row  come  to  the  same  number  as  d 
columns  with  e  times  abc  in  each  column.  It  remains 
only  to  multiply  this  number  successively  by  whatever 
factors  stand  to  the  right  of  the  interchanged  pair. 

It  follows  therefore  that  no  matter  how  many  num- 
bers are  multiplied  together,  we  may  interchange  the 
places  of  any  two  of  them  which  are  adjacent  without 
altering  the  product.  But  by  repeated  interchanges  of 
adjacent  pairs  we  may  produce  any  alteration  we  choose 
in  the  order  of  the  factors. 


ADDITION  AND  MULTIPLICATION.  37 

This  extends  the  commutative  law  of  freedom  to  all 
the  factors  in  any  product. 

Associativity.  To  show  with  equal  generality  that 
multiplication  is  associative,  we  have  only  to  prove  that 
in  any  product  any  group  of  the  successive  factors  may 
be  replaced  by  their  product. 

abcdefgh  =  abc(def}gh. 

By  the  commutative  law  we  may  arrange  the  factors 
so  that  this  group  comes  first.  Thus  abcdefgh  =  def  abc  gh. 

But  now  the  product  of  this  group  is  made  in  carry- 
ing out  the  multiplication  according  to  definition.  There- 
fore 

abcdefgh  =  def  abc  gh=  (def)  abc  gh. 

Considering  this  bracketed  product  now  as  a  single 
factor  of  the  whole  product,  it  can,  by  the  commutative 
law,  be  brought  into  any  position  among  the  other  fac- 
tors, for  example,  back  into  the  old  place;  so  abcdefgh  = 
def  abc  gh-  (def}  abc  gh  =  abc  (def)  gh. 

Distributivity.  Multiplication  combines  with  addition 
according  to  what  is  called  the  distributive  law. 

Instead  of  multiplying  a  sum  and  a  number  we  may 
multiply  each  part  of  the  sum  with  the  number  and  add 
these  partial  products. 


4x5  =  4(2+  3)  =  (2  +  3)4=2x4+3x4=5x4. 


Four  by  five  equals  five  by  four,  and 
four  rows  of  (2  +  3)  units  may  be  counted 
as  four  rows  of  two  units  together  with 
4  rows  of  3  units. 

As  the  sum  of  two  numbers  is  a  num- 


38          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

ber,   we  may  substitute    (a  +  b)    for  b  in  the  formula 
,  which  thus  gives 


So  the  distributive  law  extends  to  the  sum  of  how- 
ever many  numbers  or  terms. 

Since  a(b+c)>  ab  and  (a+b)b>ab,  there  fore  a  prod- 
uct changes  if  either  of  its  factors  changes.  A  product 
increases  if  either  of  its  factors  increases. 

Notwithstanding  the  historical  origin  of  addition 
from  counting  and  of  multiplication  from  the  addition 
of  equal  terms,  it  is  now  advantageous  to  consider  multi- 
plication, not  as  repeated  addition,  but  as  a  separate 
operation,  only  connected  with  addition  by  the  distribu- 
tive law,  an  operation  for  finding  from  two  elements, 
x,  y,  an  element  univocally  determined,  xy,  called  "the 
product,  x  by  y"  which  by  commutativity  equals  x 
times  y. 


CHAPTER  VI. 

THE  TWO  INVERSE  OPERATIONS,  SUBTRAC- 
TION AND  DIVISION. 

In  the  preceding  direct  operations,  in  addition  and 

multiplication,  the  simplest  problem  is,  from 
Inversion.  .  ,  .  ,,  .    , 

two  given  numbers  to  make  a  third. 

If  a  and  b  are  the  given  numbers,  and  x  the  unknown 
number  resulting,  then 


,  or  x  =  ax, 

according  to  the  operation. 

An  inverse  of  such  a  problem  is  where  the  result  of 
a  direct  operation  is  given  and  one  of  the  components, 
to  find  the  other  component.  The  operation  by  which 
such  a  problem  is  solved  is  called  an  inverse  operation. 

Since  by  the  commutative  law  we  are  free  to  inter- 
change the  two  parts  or  terms  of  a  given  sum,  as  also 
the  two  factors  of  a  given  product,  therefore  here  the 
inverse  operation  does  not  depend  upon  which  of  the 
two  components  is  also  given,  but  only  upon  the  direct 
operation  by  which  they  were  combined. 

Suppose  we  are  given  a  sum  which  we  designate  by 
s,  and  one  part  of  it,  say,  p,  to  find  the  corresponding 

other  part,  which,  yet  unknown,  we  repre- 
Subtraction.  _.          * 

sent  by  x.     Since  the  sum  of  the  numbers 

p  and  x  is  what  p  +  x  expresses,  we  have  the  equality 


40  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

But  this  equation  differs  in  kind  from  the  literal  equal- 
ities heretofore  used.  It  is  not  a  formula,  for  any  digital 
number  substituted  for  one  of  these  letters  restricts  the 
simultaneous  values  permissible  for  the  others.  Such  an 
equality  is  called  a  conditional  equality  or  a  synthetic 
equation,  or  simply  an  equation. 

The  inverse  problem  for  addition  now  consists  just 
in  this,  —  to  solve  the  synthetic  equation 


when  a  and  b  are  given  ;  in  other  words,  to  find  a  definite 
number  which  placed  as  value  for  x  will  satisfy  the  equa- 
tion, that  is  which  added  to  b  will  give  a,  and  thus  verify 
the  equation.  The  number  found,  which  satisfies  the 
equation  is  called  a  root  of  the  equation. 

If  the  operation  by  which  from  a  given  sum  a  and  a 
given  part  of  it  b  we  find  a  value  for  the  corresponding 
other  part  x  is  called  from  a  subtracting  b,  then,  using 
the  minus  sign  (-)  to  denote  subtraction,  we  may  write 
the  result  a-b,  read  a  minus  b. 

We  may  get  this  result,  remembering  that  a  number 
is  a  sum  of  units,  by  pairing  off  every  unit  in  b  with  a 
unit  in  a,  and  then  counting  the  unpaired  units.  This 
gives  a  number  which  added  to  b  makes  a. 

The  expression  or  result  a-  b  is  called  a  difference. 

The  term  preceded  by  the  minus  sign  is  called  the 
subtrahend',  the  other  the  minuend. 

Thus  (a-b)+b  =  a-b  +  b  =  a;  also 


Ordinally,  to  subtract  y  from  x  is  to  find  the  number 
occupying  the  ;yth  place  before  x. 

Postulating  the  "rule  of  signs,"  that  a-(b-c)  = 
a-b  +  c,  subtraction  is  associative  and  commutative. 


SUBTRACTION  AND  DIVISION.  41 

The  term  division  has  two  distinct  meanings  in  ele- 
mentary mathematics.     There  are  two  ope- 
Division.  .  ...  .  .    . 

rations  called  division:  1°,  Remainder  divi- 
sion ;  2°,  Multiplication's  inverse. 

1°,  Given  two  numbers,  a  >  b,  a  the  dividend,  and  b 
the  divisor,  the  aim  of  remainder  division  may  be  con- 
sidered the  putting  of  a  under  the  form  bq  +  r,  where 
r  <  b,  and  b  not  0.  We  call  q  the  quotient,  and  r  the  re- 
mainder. Both  are  integral.  There  is  a  definite  proba- 
bility that  r  will  not  be  0. 

The  remainder  division  of  a  by  &  answers  the  two 
questions:  1°,  What  multiple  of  b  if  subtracted  from  a 
gives  a  difference  or  remainder  less  than  6?  2°,  What 
is  this  remainder? 

Remainder  division  will  regroup  a  given  set,  the  divi- 
dend, into  smaller  sets  each  with  the  same  cardinal  as  a 
given  set,  the  divisor,  and  a  remaining  set  whose  cardinal 
is  less  than  that  of  the  divisor. 

The  number  of  the  equivalent  subsets  is  here  the 
quotient.  There  is  no  implication  that  the  original  units 
are  equal  in  size.  So  it  would  be  a  blunder  to  call  this 
process  measuring. 

Again  remainder  division  will  regroup  a  given  set, 
the  dividend,  into  equivalent  subsets  and  a  less  remainder, 
when  the  number  of  subsets,  the  divisor,  is  given.  The 
cardinal  of  each  subset  is  here  the  quotient.  This  has 
sometimes  been  called  partitive  division.  But  these  two 
applications  of  remainder  division  are  not  two  kinds  of 
division,  and  should  not  be  emphasized.  In  arithmetical 
division,  dividend  and  divisor  are  two  given  numbers 
fixing  a  third,  the  quotient.  So  the  division  of  15  by  4 
tells  how  often  15  eggs  contain  4  eggs  and  equally  well 


42          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

how  many  dollars  in  each  of  the  4  equivalent  pieces  of 
15  dollars. 

When  r  is  0,  then  a  is  a  multiple  of  b,  and  a  is  exactly 
divisible  by  b. 

The  case  6  =  0  is  excluded.  In  this  excluded  case  the 
problem  would  be  impossible  if  a  were  not  0.  But  if 
a-0  and  b  =  0,  every  number,  q,  would  satisfy  the  equal- 
ity a  =  bq.  So  this  case  must  be  excluded  to  make  the 
operation  of  division  unequivocal,  that  is,  in  order  that 
the  problem  of  division  shall  have  always  one  and  only 
one  solution.  A  second  solution  q',  /  would  give  a  =  bq  +  r 
=  bq'+r*t  b(q-q')=r'-r.  But  r/-r<b,  while  b(q-q') 
not  <  b. 

2°,  Division  may  also  be  regarded  as  the  inverse  of 
multiplication.  Its  aim  is  then  considered  to  be  the  find- 
ing of  a  number  q  (quotient)  which  mulitplied  by  b  (the 
divisor)  gives  a  (the  dividend).  Here  division  is  the 
process  of  rinding  one  of  two  factors  when  their  product 
and  the  other  factor  are  given. 

The  result  q  is  represented  by  a/b.  If  a=0,  then 
g  =  0.  This  definition  of  division  gives  the  equality 


Remember  b  ^  0,  that  is,  b  not  equal  to  0. 
In  particular  a/1  =a. 

Postulating  the  rule  a/(b/c)  =a/bxc,  division  is  as- 
sociative, commutative,  and  distributive. 

(a  +  b)/c=(a/c)+(b/c)  ;  but 


In  general     1° 

2°  (a-b)/m  =  a/m-b/m. 


SUBTRACTION  AND  DIVISION.  43 

3°  a(b/c)=ab/c. 


6°  a/b  =  am/bm. 

7o  a/b=(a/m)/(b/m). 

The  Arabs,  as  early  as  1000  A.  D.,  used  the  solidus, 
or  slant-sect  /  and  also  the  horizontal  sect,  as  in  $  or  %,  to 
denote  the  quotient  of  the  first  or  upper  number  by  the 
other. 

The  symbol  -f-  is  not  found  until  about  1630.  It 
may  have  been  suggested  by  the  use  of  the  horizontal 

sect  in  ~.    Turned  on  end  •!•  I  use  it  for  symmetrical, 
as  in  •!•  A  for  isosceles  triangle. 


CHAPTER  VII. 
TECHNIC. 

In  adding  a  column  of  digits,  consider  two  numbers 

together,  but  only  think  their  sum. 

Addition.  °XT         -it-  ,i_-  i 

Now   in  adding  up    this  column    only 

think  9,  16,  18,  27,  32,  43,  stressing  forty, 

3  23     and  writing  down  the  three  while  think- 

8  48     ing  it. 

5  35  The  stress  on  the  forty  is  to  hold  the 

9  59  four  in  mind  for  use  in  the  next  column 
2  62  to  the  left.     Such  a  number  is  said  to  be 
7  87  carried.    Begin  adding  up  the  next  column 

4  74  to  the  left  by  thinking  13. 

5  95  To   check   the   work,   add   the   column 


43  3     downward,  since  mere  repetition  of  work 

tends  to  repeat  the  mistake  also. 

Look  at  the  question  of  subtracting  as  asking  what 

number  added  to  the  subtrahend  gives  the 
Subtraction.  ,       .  ,  .       , 

minuend.    Always  work  subtraction  by  add- 
ing.    Thus  subtract  1978  from  3139  as  follows:  Think 
8  and  one  make  9;  7  and  six  make  13,  carry 
3139       1;  10  and  one  make  11,  carry  1;  2  and  one 
1978       make  3.     Write  down  the  spelled  digits  just 
1161       while  thinking  them. 
Explain  "carrying"  by  the  principle  that  the  difference 


TECHNIC.  45 

between  two  numbers  remains  the  same  though  they  be 
given  equal  increments. 

9254  Again  think,  5  and  nine  make  14,  carry  1 ; 

8365       7  and  eight  make  1 5,  carry  1 ;  4  and  eight  make 
889       12,  carry  1 ;  9  equals  9. 

In  working  the  examples  we  have  added 
downwards,  so  check  by  adding  upwards  the  difference 
(the  answer)  to  the  subtrahend;  think  (for  9  and  5)  14, 
(for  9  and  6)  15,  (for  9  and  3)  12,  (for  1  and  8)  9. 

It  is  preferable  for  several  reasons  to  perform  numer- 
ical operations  from  the  left.  An  operation  thus  cor- 
responds more  closely  with  the  process  it  represents. 
Again  this  way  fixes  the  attention  at  once  upon  the 
greater,  more  important  parts  of  the  quantities  concerned, 
permitting  immediate  approximations,  and  so  giving  speed 
in  dealing  with  life  realities,  thus  increasing  practical 
efficiency. 

Though  the  immediate  conception  of  a  large  multiple 
of  a  small  number,  perhaps  because  of  our  mastery  of 
the  number  series,  is  simpler  than  that  of  a  small  multiple 
of  a  large  number,  yet  operatively,  as  a  multiplication, 
the  latter  gives  the  easier  process.  Hence  choose  the 
smaller  as  your  multiplier.  To  find  thrice  2104  it  would 
be  best  to  apply  the  distributive  law  from  the  left,  giving 
3(2000  +  100  +  4).  This  is  the  way  of  the  lightning  cal- 
culator. Meantime,  as  a  concession,  we  teach  the  back- 
ward application  of  the  law,  (2000+100  +  4)3. 

Set  down  the  multiplier  precisely  in  column  under 
Multiplica-  the  multiplicand,  units  under  units.  Begin 
tion.  by  multiplying  the  units  figure  of  the  mul- 

tiplicand by  the  leftmost  figure  of  the  multiplier,  writing 
under  this  leftmost  figure  the  first  figure  thus  obtained. 


46          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

35427       Then  use  the  successive  figures  in  order. 
1324       The  figure  set  down  from  multiplying  the 


35427  units  always  comes  precisely  under  its  mul- 

106281  tiplier. 

70854  The  advantage  of  this  method  is  that 

141708  it  gives  the  most  important  partial  product 


46905348  first,  and  in  abridged  or  approximate  work 
one  or  two  of  the  leftmost  figures  may  be 
all  that  are  wanted. 

Rule :  If  of  two  figures  multiplied  one  is  in  units  col- 
umn, the  figure  set  down  stands  under  the  other. 

Check  by  casting  out  nines. 

Proceed  as  follows :  Add  the  single  figures  of  the 
multiplicand,  but  always  diminish  the  partial  sums  by 
Verify  multi-  dropping  nine.  The  remainder  is  identical 
plication.  with  the  remainder  found  much  more  labori- 
ously by  dividing  by  nine.  Thus  35427  gives  3,  since  7 
and  2  give  nine  as  also  4  and  5.  Find  just  so  the  remain- 
der of  the  multiplier.  Here  1324  gives  1.  If  our  work 
is  correct,  the  remainder,  or  excess,  of  the  product  of 
these  two  remainders  equals  the  remainder,  or  excess, 
for  our  product.  Here  46905348  gives  3. 

The  complete  proof  of  this  method  of  verification 
lies  simply  in  the  fact  that  the  remainder  left  when  any 
number  is  divided  by  nine  is  the  same  as  that  left  when 
the  sum  of  its  digits  is  divided  by  nine.  For  10-1=9, 
1 00  - 1  =  99,  1 000  - 1  =  999,  etc.  Hence  i  f  f  rom  any  num- 
ber be  taken  its  units,  also  a  unit  for  each  of  its  tens,  a 
unit  for  each  of  its  hundreds,  a  unit  for  each  of  its  thou- 
sands, etc.,  the  remainder  is  a  multiple  of  nine.  But  the 
part  taken  away  is  the  sum  of  the  number's  digits. 


TECHNIC. 


47 


Shorter 
forms. 


(a)  When  the  multiplier  contains  only  two  digits, 
shorten  the  work  by  adding  in  the  results 
of  the  multiplication  by  the  second  digit 
to  that  already  obtained.  Here,  after  multi- 
plying by  3,  think  fourteen;  16,  17  eighteen', 
10,  11,  seventeen;  18,  19,  twenty-six;  ten; 
three.  Write  down  the  unaccented  part  of 
these  spelled  numbers  while  thinking  it. 


9587 
32 


28761 


306784 


9867  (b)  If  in  a  multiplier  of  only  two  digits 

15  either  is  unity,  write  only  the  answer. 

148005  Here  think  thirty-five ;  30,  33,  forty ;  40, 

44,  fifty;  45,  50,  fifty-eight;  fourteen. 

7968  Here  think  eight;  32,  thirty-tight;  24, 

41  27,  thirty-six;  36,  39,  forty-six;  28,  thirty- 

326688  two. 

(c)  When  in  a  three -place  multiplier 
1234       taking  away  either  end-digit  leaves  a  mul- 

568       tiple  of  it,  shorten  by  adding  to  the  digit's 

9872       partial  product  the  proper  multiple  of  it. 
69104  After  multiplying  by  8,   multiply  this 

700912       partial  product  by  7  (tens). 

4213 

864  After  multiplying  by  the  8,  (hundreds), 

33704  multiply  this  partial  product  by  8.     This 

269632       gives  units. 
3640032 

(Divisor  an  integer)  : 

Write  the  first  figure  of  the  quotient  precisely  over 
the  last  figure  of  the  first  partial  dividend. 
Use  no  bar  to  separate  them. 


Division. 


48          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

Omit  the  partial  products,  the  multiples  of  the  divisor, 
writing  down  the  differences  while  doing  the  multiplica- 
tion. 

318  Nineteen  into  60  thrice.    Three  nines 

19)  6054       are  27,  and  three  makes  30.     Carry  3. 

3  Three  ones  are  3;  say  6. 

16  Nineteen  into  35  once.     One  nine  is 

12       9,  and  six  makes  15.    Carry  1.    One  1  is 

1 ;  say  2,  and  one  makes  3. 

Nineteen  into  164  eight  times.  Eight  nines  are  72, 
and  two  makes  74.  Carry  7.  Eight  ones  are  8;  say  15, 
and  one  makes  16. 

Here,  using  the  2,  think  16  and  naught,  1'6.     Carry 

1.     10,  say  11  and  six,   1'7.     Carry  1. 

27       6,  say  7  and  two,  9.     Thus  we  get  the 

358)  9762       new  partial  dividend  2602,  which  gives 

260         in  our  quotient  7.     Using  this  7,  think 

96       56  and  six,  6'2.     Carry  6.     35,  say  41 

and    nine,   5'0.     Carry  5.     21,   say  26. 

Thus  we  get  our  remainder  96. 

This  method  gives  at  once  the  true  value  of  each 
partial  quotient.  Moreover  its  using  the  partial  products 
instead  of  setting  them  down,  actually  diminishes  error, 
besides  being  easier  and  quicker  and  more  compact. 

The  excess  of  the  product  of  excesses  of  divisor  and 
Verify  quotient  increased  by  excess  of  remainder 

division.          equals  excess  of  dividend. 

In  our  example  the  excess  from  the  quotient  is  0. 
So  the  excess  from  the  dividend,  6,  equals  that  from 
the  remainder. 


CHAPTER  VIII. 
DECIMALS. 

A  decimal  is  a  number  whose  expression  in  our  posi- 
tional notation  contains  digits  to  the  right  of  units  col- 
umn.    A  decimal   is  a  basal  subunital;  a 
Decimals.  .    .  . 

number  containing  subunits  which  are  mul- 
tiples of  minus  powers  of  the  base. 

It  is  the  characteristic  of  our  positional  notation  for 
number  that  shifting  a  digit  one  place  to  the  left  multi- 
plies it  by  the  base  of  the  system.  The  zero  enables  us  to 
indicate  such  shifting.  Thus  since  our  base  is  ten,  1 
shifted  one  place  to  the  left,  10,  becomes  ten;  two  shifted 
two  places  to  the  left,  200,  becomes  two  hundred. 

Inversely,  shifting  a  digit  one  place  to  the  right,  di- 
vides it  by  the  base  of  the  system.  Thus  3  in  the  thou- 
sands place,,  3000,  shifted  one  place  to  the  right  becomes 
300. 

We  now  create  that  this  shifting  to  the  right  may  go 
on  beyond  the  units'  place  with  no  change  of  meaning  or 
effect. 

In  order  to  write  this,  we  use  a  device,  a  notation  to 
mark  or  point  out  the  units  place,  a  point  immediately 
to  its  right  called  the  decimal  point  or  unital  point.  Our 
present  decimal  notation,  a  development  of  that  of  Simon 
Stevinus  of  Bruges,  1585,  was  not  generally  used  before 
the  eighteenth  century,  although  the  decimal  point  appears 
first  in  1617  on  page  21  of  Napier's  Rabdologiae.  Thus 


50          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

4  shifted  one  place  to  the  right  becomes  0.4  and  of  course 
means  a  number  which  multiplied  by  the  base  gives  4. 
Such  numbers  have  been  called  decimals.  Their  theory 
is  independent  of  the  base,  which  might  be  say  12  or  2, 
in  which  case  the  word  decimals  would  be  a  distinct  mis- 
nomer. 

The  perfection  of  our  system  is  in  its  subtle  use  of  a 
base-number,  not  in  that  number  ten.  Our  system  is  a 
miraculous  instrument  for  easy  reckoning,  not  because 
it  is  decimal,  but  because  the  digit  figures,  by  aid  of  a 
potential  zero,  always  express,  in  their  orderly  position, 
to  left  or  right  of  a  point,  multiples  of  ascending  and 
descending  powers  of  one  basal  number.  Thus  9(10)4  + 


3(10)-3  =  90876.543. 

If  however  the  base  be  ten,  then  shifting  a  digit  one 
place  to  the  left  multiplies  it  by  ten.  But  this  is  accom- 
plished for  every  digit  in  the  number  simply  by  shifting 
the  point  one  place  to  the  right.  Thus  .05  is  tenfold 
.005.  If  our  unit  is  a  dollar,  $1,  then  the  first  place  to 
the  right  will  be  dimes.  Thus  $0.6  means  six  dimes. 
The  next  place  to  the  right  of  dimes  means  cents.  Thus 
$.07  means  seven  cents.  The  next  place  to  the  right 
of  cents  means  mills.  Thus  $.008  means  eight  mills. 

Ten  mills  make  a  cent.  Ten  cents  make  a  dime.  Ten 
dimes  make  a  dollar. 

In  general  we  name  these  basal  subunitals  so  as  to 
indicate  by  symmetry  their  place  with  reference  to  the 
units'  column.  As  the  first  column  to  the  left  of  units  is 
tens,  so  the  first  column  to  the  right  of  units  is  called 
tenths.  As  the  second  column  to  the  left  of  the  units' 
column  is  called  hundreds,  so  the  second  column  to  the 
right  of  the  units'  column  is  called  hundredths.  As  the 


• 

DECIMALS.  51 

third  column  to  the  left  of  the  units'  column  is  called 
thousands,  so  the  third  column  to  the  right  of  the  units' 
column  is  called  thousandths. 

But  these  names  need  not  be  used  in  reading  a  sub- 
unital.  Thus  0.987  may  be  read:  Point,  nine,  eight, 
seven.  So  mathematicians  read  it,  and  all  educated  sci- 
entists. 

One-tenth  is  a  number  ten  of  which  are  together 
equal  to  a  unit.  "Point,  one,"  says  this. 

If  an  integer  be  read  by  merely  pronouncing  in  succes- 
sion the  names  of  its  digits,  as  in  reading  7689  as  seven, 
six,  eight,  nine,  we  do  not  know  the  rank  and  so  all  the 
value  of  any  figure  read  until  after  all  have  been  read. 

Hence  the  advantage  of  reading  7689  seven  thousand 
six  hundred  and  eighty-nine.  But  in  reading  the  decimal 
.7689  as  "point,  seven,  six,  eight,  nine"  we  know  every 
thing  about  each  figure  as  it  is  read,  which  on  the  con- 
trary we  do  not  know  if  it  be  read  seven  thousand  six 
hundred  and  eighty-nine  ten-thousandths. 

Morever  such  a  habit  of  reading  decimals  detracts 
from  our  confident  certainty  of  understanding  integers 
step  by  step  as  read.  There  may  be  coming  at  the  end  a 
wretched  subunital  designation  like  this  "ten-thousandths" 
to  metamorphose  everything  read. 

So  always  read  decimals  by  pronouncing  the  word 
point  and  the  names  of  the  separate  single  digits. 

Read  700.008  seven  hundred,  point,  naught,  naught, 
eight  Read  .708  point,  seven,  naught,  eight. 

This  wholly  obviates  the  imaginary  difficulty  of  the 
hysterical  country  school  ma'am  (unmarried),  whose  hy- 
pothetical man  she  supposed  could  not  properly  inflect  his 
voice,  and  so  could  not  by  tone  indicate  the  difference 
marked  by  punctuation,  between  "seven  hundred,  and 


52  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

eight  thousandths,"  and  "seven-hundred-and-eight  thou- 
sandths." To  relieve  her  wooden  man,  her  femininity 
suggested  the  crime  of  suppressing  the  "and"  in  all  such 
good  English  phrases  as  The  Thousand  and  One  Nights. 


-4  =  9876.  5432. 

To  add  decimals,  write  the  terms  so  that  the  decimal 
points  fall  precisely  under  one  another,  in  a  vertical  col- 
umn. Then  proceed  just  as  with  integers,  the  point  in  the 
sum  falling  under  those  of  the  terms. 

Just  so  it  is  with  subtraction. 

In  multiplying  decimals   remember  we  are  dealing 

simply  with  a  symmetrical  completion,  ex- 

tension of  positional  notation  to  the  right 

from  units'  place.     Realize  the  perfect  balance  resting 

on  the  units'  column.    4321  .234. 

A  shift  of  the  decimal  point  changes  the  rank  of  each 
of  the  digits.  So  to  multiply  or  divide  by  any  power  of 
ten  is  accomplished  by  a  simple  shift  of  the  point. 

Thus  98  .  76  x  1  0  is  987  .  6.  Just  so  98  .  76/10  is  9  .  876, 
and  is  identical  with  98.76x0.1.  Twice  this  is  98.76x 
0.1x2  or  98.76x0.2=19.752. 

So  to  multiply  by  a  decimal  is  to  multiply  by  an  in- 
teger and  shift  the  point. 

Hence  the  rule,  useful  for  check,  that  the  number 
of  decimal  places  in  the  product  is  the  sum  of  the  places 
in  the  factors.  There  is  no  need  for  thinking  of  tenths 
as  fractions  to  realize  that  two-tenths  of  a  number  is 
twice  one-tenth  of  it. 

In  multiplying  decimals,  write  the  multiplier  so  that 
its  point  comes  precisely  under  the  point  in  the  multipli- 
cand, and  in  vertical  column  with  these  put  the  point  in 


DECIMALS.  53 

each  partial  product.     The  figure  obtained  from  multi- 
plying the  units  figure  of  the  multiplicand  must  come 
precisely  under  the  figure  by  which 
1293.015  we  are  multiplying. 

132.02  Here,  beginning  to  multiply  by  the 

129301.5  1,   think   five  while  writing    it    two 

38790.45  places  to  the  left  of  the  figure  multi- 

2586.030          plied  because  the  1  is  two  places  to 

25.86030       the  left  of  the  units' column.  Proceed 


170703.8403  to  multiply  by  the  3,  thinking  fifteen; 
3,  four;  naught;  nine;  twenty-seven; 
etc. 

Rule :  Multiplying  shifts  as  many  places  right  or  left 
as  the  multiplier  is  from  the  units'  column. 

Here  think  twenty-one  while  writ- 
41.27  ing  the   1    two   places   more   to  the 

.03  right  than  the  7  because  the  3  is  two 

1 . 2381         places  to  the  right  of  the  units'  col- 
umn. 
In  division  of  decimals  place  the  decimal  point  of 

the  quotient  precisely  over  the  decimal  point 
Quotient.  -    *  r  . 

of  the  dividend  and,  when  the  divisor  is  an 

integer,  the  first  figure  of  the  quotient  over  the  last  figure 
of  the  first  partial  dividend. 

Rule:  The  first  figure  of  the  quo- 

638  tient  stands  as  many  places  to  the  left 

.021)13.4         of  the  last  figure  of  the  first  partial  divi- 

8         dend  as  there  are  decimal  places  in  the 

17       divisor. 

2  Here  the  quotient  638  is  an  integer. 

The  sign  +  at  the  end  of  a  number  means  there  is 

a  remainder,  or  that  the  number  to  which  it  is  attached 


54          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

6+       falls   short   of   completely,   exactly  ex- 
2 . 1 )  .  01 34       pressing  all  it  represents,  though  increas- 
8       ing  the  last  figure  by  unity  would  over- 
pass exactitude  and  so  should  be  fol- 
lowed by  the  sign  -  (minus). 

Thus  :r  =  3.14  +  and 

77  =  3.1416- 

This  is  historically  the  first  meaning  of  the  signs  + 
and  -,  which  arose  from  the  marks  chalked  on  chests  of 
goods  in  German  warehouses,  to  denote  excess  or  defect 
from  some  standard  weight. 

When  there  is  a  remainder  we  may  get  additional 

places  in  the  quotient  by  annexing  ciphers 

63  to  the  dividend  and  continuing  the  division. 

.21)13.4  The  phrase   "true  to  2    (or  3,   etc.) 

8         places  of  decimals"  means  that  a  closer 

17       approximation  can  not  be  written  without 

using  more  places. 

Thus  as  a  value  for  TT,  3.14  is  true  or  "correct"  to 
two  places  of  decimals,  since  7r  =  3.14159  +  ;  while  3. 1416 
is  true  to  four  places. 

As  an  approximation  to  1.235  we  may  say  either 
1 .23  or  1 .24  is  true  to  two  places  of  decimals. 


CHAPTER  IX. 
FRACTIONS. 

Generality  is  the  essence  of  modern  mathematics. 
The  creative  extension  of  its  previously  attained  system 

_         ,.          marks  the  growth  of  its  powers  as  our  great- 
Generahza-  f      ,. 

tions  of  est  instrument  for  that  ordering  and  simpli- 
fication of  our  universe,  that  transforming 
of  chaos  into  cosmos,  which  is  the  vocation  of  science. 
Such  an  extension  of  the  original  integral  numbers  we 
see  in  decimals.  But  just  here  we  have  one  of  the  sharp 
rebuttals  found  everywhere  in  mathematics  to  the  peda- 
gogic principle  that  education  should  recapitulate  the  path 
of  the  race.  Decimals,  roughly  two  centuries  old,  should 
be  taught  before  fractions,  probably  more  than  five  thou- 
sand years  old.  The  romantic  treatise  we  still  possess, 
entitled  "Directions  for  obtaining  the  Knowledge  of  all 
Dark  Things,"  written  by  the  scribe  Ahmes  about  1700 
B.  C,  and  founded  on  an  older  work  believed  by  the 
Egyptologist  Birch  to  date  back  as  far  as  3400  B.  C., 
contains,  to  solve  the  problem  of  representing  any  frac- 
tion as  a  sum  of  fractions  each  with  numerator  one,  a 
table  of  solutions  for  all  fractions  with  numerator  2  and 
all  denominators  from  3  to  99;  e.  g.,  2/99  =  1/66+1/198. 
Expansions  of  the  number-idea  are  guided  by  one 
criterion,  that  there  be  no  break  in  the  applicability  of 
the  old  formal  conventions  of  procedure.  They  are  mo- 
tived by  the  desire  to  obviate  exceptions.  Thus  after 


56          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

decimals  and  fractions  or  rationals,  mathematicians  cre- 
ated reals,  and  signed  numbers,  and  complex  numbers. 

For  the  new  numbers  hold  the  old  laws. 

1  st.  Every  number  combination  which  gives  no  already 
Principle  of  existing  number,  is  to  be  given  such  an 
permanence,  interpretation  that  the  combination  can  be 
handled  according  to  the  same  rules  as  the  previously 
existing  numbers. 

2d.  Such  combination  is  to  be  defined  as  a  number, 
thus  enlarging  the  number  idea. 

3d.  Then  the  usual  laws  (freedoms)  are  to  be  proved 
to  hold  for  it. 

4th.  Equal,  greater,  less  are  to  be  defined  in  the  en- 
larged domain. 

This  was  first  given  by  Hankel  as  generalization  of 
a  principle  given  by  G.  Peacock,  British  Association,  III, 
London,  1834,  p.  195.  Symbolic  Algebra,  Cambridge, 
1830,  p.  105;  2d  ed.,  1845,  p.  59. 

If  unity  in  pure  number  be  considered  as  indivisible, 
fractions  may  be  introduced  by  conventions.  Take  two 

integers  in  a  given  order  and  regard  them 
Fractions.  * 

as  forming  a  couple  with  sense;  create  that 

this  ordered  couple  shall  be  a  number  of  a  new  kind,  and 
define  the  equality,  addition,  and  multiplication  of  such 
numbers  by  the  conventions, 

a/b  =  c/d  if  ad  =  bc; 

a/b  +  c/d=  (ad  +  bc}/bd; 


The  preceding  number  is  called  the  numerator  of  the 
fraction  ;  the  succeeding  number,  the  denominator. 

Fractions  have  application  only  to  objects  :apable  of 
partition  into  equal  portions  equal  in  number  to  the  de- 
nominator. No  fraction  is  applicable  to  a  person. 


FRACTIONS.  57 

In  accordance  with  the  principle  of  permanence,  we 
create  that  the  compound  symbol  of  the  form  a/b,  two 
natural  numbers  separated  by  the  slant,  shall  designate 
a  number.  Either  the  symbol  or  the  number  may  be 
called  a  fraction.  The  slant  is  to  stand  for  the  division 
of  a  by  b,  of  the  preceding  by  the  succeeding  number, 
where  this  is  possible.  When  a  is  exactly  divisible  by  b, 
that  is,  without  remainder,  the  fraction  designates  a  nat- 
ural number.  Always  notationally  a  fraction  represents 
an  unperformed  operation,  a  division,  and  any  approxi- 
mate result  of  the  performance  of  this  division  is  an 
approximate  value  of  the  fraction ;  but  the  number  repre- 
sented by  the  fraction  is  always  exact,  precise,  definite, 
perfect. 

When  a  is  a  multiple  of  b,  and  a'  of  b',  the  equality 
ab'=a'b  is  the  necessary  and  sufficient  condition  for  the 
symbols  a/b,  a'/b'  to  represent  the  same  number.  By  this 
same  condition  we  define  the  equality  of  the  new  numbers, 
the  fractions. 

A  fraction  is  irreducible  when  its  numerator  and  de- 
nominator contain  no  common  factor  other  than  1. 

To  compare  two  fractions,  reduce  them  to  a  common 
denominator,  then  that  which  has  the  greater  numerator 
is  called  the  greater. 

A  proper  fraction  is  a  fraction  with  numerator  less 
than  denominator.  It  is  less  than  1. 

Subtraction  is  given  by  the  equality  a/b -a' '/&'=  (abf- 
a'b')/bb'. 

The  multiplication  of  fractions  is  covered  by  the 
statement:  A  product  is  the  number  related  to  the  mul- 
tiplicand as  the  multiplier  is  to  unity. 

(a/b)  (a'/b')=aa'/bb'. 


58          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

Thus  ( 5/7) x  (2/3)  means  trisect,  take  one  of  these 
three  parts,  then  double,  giving  10/21. 

So  (a/b}x(b/a)  =  1.  Two  numbers  whose  product 
is  unity  are  called  reciprocal. 

Extending  the  meaning  of  "times"  so  that  2/3  times 
thrice  equals  twice,  and  n/d  times  d  times  equals  n  times, 
we  have  {x/z)z=x.  Hence  5/7  times  Q  is  a  quantity  such 
that  7  times  it  gives  5Q.  Therefore  it  equals  5  times  a 
quantity  seven  of  which  make  Q,  that  is  five-sevenths 

ofQ.  ' 

So  2/3  times  Q  is  2/3  of  Q. 

Division  is  taken  as  the  inverse  of  multiplication, 
hence  (c/d)/(a/b)  means  to  find  a  number  whose  prod- 
uct with  (a/b)  is  (c/d).  Such  is  (c/d)(b/a). 

So  (c/d)/(a/b)  =  (c/d)(b/a")=bc/ad. 

1°.  This  last  expression  may  be  considered  simply  a 
more  compact  form  of  the  first,  obtained  by  reducing 
to  a  common  denominator  and  cancelling  this  denom- 
inator. This  compact  form  can  be  obtained  by  a  proce- 
dure sometimes  called  the  rule  for  division  by  a  fraction : 
Invert  the  divisor  and  multiply., 

2°.  If  we  interchange  numerator  and  denominator 
of  a  fraction  we  get  its  inverse  or  reciprocal.  So  the 
inverse  of  a  is  I/a. 

(a/6)  (&/«)=!- 

Now  (x/y}/(a/b}  means  to  find  a  number  which 
multiplied  by  a/b  gives  x/y,  and  so  the  answer  is 
(x/y)(b/a).  Hence:  To  divide  by  a  fraction,  multiply 
by  its  reciprocal. 

3°.  Again  to  find  (a/b}/(c/d),  note  that  c/d  is  con- 
tained in  1  d/c  times,  and  hence  in  a/b  it  is  contained 
(a/b)(d/c)  times. 


FRACTIONS.  59 

A  reduced  fraction  is  one  whose  numerator  and  de- 
Fractions        nominator  contain  no  common  factor. 
ordered.  j^  f  ractions  arranged  according  to  size 

are  an  ordered  set,  but  not  well  ordered;  for  no  fraction 
has  a  determinate  next  greater  fraction,  since  between 
any  two  numbers,  however  near  in  size,  lie  always  in- 
numerable others. 

But  all  reduced  fractions  can  be  arranged  in  a  well- 
ordered  set  arranged  according  to  groups  in  which  the 
sum  of  numerator  and  denominator  is  the  same: 

1/1,  1/2,  2/1,  1/3,  3/1,  1/4,  2/3,  3/2,  4/1,  1/5, 
5/1,  1/6,  2/5,  3/4,  4/3,  5/2,  6/1,. . . . 

Thus  they  make  a  simply  infinite  series  equivalent  to 
the  number  series. 

Proper  fractions  can  be  arranged  by  denominators: 


To  turn  a  fraction  a/b  into  a  decimal  c/\0k  must 
give  a\Qk  =  bc,  where  c  is  a  whole  number.  Since  a/b 
is  in  reduced  form,  therefore  a  and  b  have  no  common 
factor.  So  10^  must  be  exactly  divisible  by  b.  Thus 
only  fractions  with  denominator  of  the  form  2n5m  can 
be  turned  exactly  into  decimals. 

Fractions  may  be  thought  of  as  like  decimals  in  be- 
ing also  subunitals.  The  unit  operated  with  in  a  fraction, 
the  fractional  unit,  is  a  subunit,  and  the  denominator  is  to 
tell  us  just  what  subunit,  just  what  certain  part  of  the 
whole  or  original  or  primal  unit  is  taken  as  this  subunit ; 
while  the  numerator  is  the  number  of  these  subunits.  The 
denominator  tells  the  scale  of  the  subunit,  its  relation 
to  the  primary  integral  unit.  Thus  3/10  is  a  three  of 
subunits  ten  of  which  make  a  unit.  Thus,  like  an  integer, 


60  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

a  fraction  is  a  unity  of  units  (or  one  unit),  but  these 
are  subunits  Different  subunits  may  be  very  simply 
related,  as  are  1/2,  1/4,  1/8. 

To  add  3/4  and  1/2  we  first  make  their  subunits  the 
same  by  bisecting  the  subunit  of  1/2,  which  thus  be- 
comes 2/4.  Then  3/4  and  2/4  may  be  counted  together 
to  give  5/4. 

Fractions  having  the  same  subunit  are  added  by  add- 
ing their  numerators,  the  same  denominator  being  re- 
tained since  the  subunit  is  unchanged.  The  like  is  true 
of  subtraction. 

To  add  unlike  fractions  change  to  one  same  subunit. 
The  technical  expression  for  this  is  "reduce  to  a  common 
denominator." 

Since  we  already  know  that  to  be  counted  together 
the  things  must  be  taken  as  indistinguishably  equivalent, 
the  procedure  of  changing  to  one  same  subunit  is  crystal 
clear. 

To  change  a  half  to  twelfths  is  simply  to  split  up  the 
one-half,  the  first  subunit,  into  subunits  twelve  of  which 
make  the  whole  or  original  unit. 

Thus,  operatively,  to  express  a  fraction  in  terms  of 
some  other  subunit,  the  procedure  is  simply  to  multiply 
(or  divide)  numerator  and  denominator  by  the  same 
number. 

Thus  1/2=  (Ix6)/(2x6)  =6/12. 

So  6/12=  (6/3)/(12/3)  =2/4. 

This  principle  in  the  form :  "The  value  of  a  fraction 
is  unaltered  by  dividing  both  numerator  and  denominator 
by  the  same  number,"  is  freely  applied  in  what  is  tech- 
nically called  "reducing  fractions  to  their  lowest  terms." 

It  should  be  applied  just  as  freely  and  directly  in  the 
form :  "The  value  of  a  fraction  is  unaltered  by  multiply- 


FRACTIONS.  61 

ing  both  numerator  and  denominator  by  the  same  num- 
ber." Thus  the  complex  fraction  (2  +  2/3)/(3  +  2/9), 
multiplying  both  terms  by  9,  gives  at  once  24/29.  Again 
(3  feet  5  inches)/ (2  feet  7  inches),  multiplying  both 
terms  by  12,  gives  41/31. 

13%  To  subtract  7  +  3/4  from  13  +  1/4,  that  is 

7%       to  evaluate   13% -7%,   think  3/4    and    two- 

5%       fourths  make  5/4,  carry  1 ;  8  and  five  make 
thirteen. 

The  1  in  l/n  is  the  suounit,  the  n  specifying  what 
particular  subunit.  In  division  of  a  fraction  by  an  in- 
Division  of  teger  we  meet  the  s^me  limitation  which 
fractions.  theoretically  led  to  the  creation  of  fractions ; 
namely  2/5  is  no  more  divisible  by  three  than  any  other 
two.  But  here  we  can  easily  transform  our  fraction  into 
an  equivalent  divisible  by  3.  Just  trisect  the  subunit. 
Thus  2/5  becomes  6/15,  which  is  divisible  by  3  giving 
2/15. 

Such  result  is  always  at  once  attained  simply  by  mul- 
tiplying the  given  denominator  by  the  given  integral 
divisor.  Hence  the  rule:  To  divide  a  fraction  by  an 
integer,  multiply  its  denominator  by  the  integer. 

Our  multiplication  is  to  be  associative,  so  when  the 
multiplier  is  increased  any  whole  number  of  times,  the 
M  .  .  ...  product  will  be  increased  the  same  number 
tion  of  of  times.  For  instance,  thrice  5  is  15. 

Doubling  the  multiplier,  twice  thrice  5  is 
30,  which  is  double  the  former  product  15.  So  for  frac- 
tions, as  4/7  and  9/10,  the  product  is  such  that  when 
the  multiplier  4/7  is  increased  7  times,  so  is  the  product. 
Now  7  times  4/7  is  4.  Thus  4  times  the  fraction  9/10 
will  be  7  times  the  required  product.  But  4  times  9/10 
is  36/10,  and  the  seventh  part  of  this  is  36/70.  Let 


62  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

this  then  be  our  product  of  4/7  and  9/10.  We  reach 
thus  for  the  product  of  two  fractions  the  rule :  Multiply 
the  numerators  together  for  the  new  numerator,  and  the 
denominators  for  the  new  denominator. 


CHAPTER  X. 
RELATION  OF  DECIMALS  TO  FRACTIONS. 

6.214  Fractions  may  be  freely  combined  with 

3%  decimals.     Thus  1/24=  .04%. 

18.642  1  meter  =  39.37  inches  =  3  feet  3% 

2.071%      inches. 

20.713%  In   finding  the  product  of   a  decimal 

and  a  fraction  use  the  fraction  as  multiplier. 

By  our  positional  notation,  0 . 1  means  one  subunit 

,  «.  r>    •     ,    such  that  ten  of  them  make  the  unit.    But 

1st.  Decimals 

into  just   this  same  thing    is  meant    by  1/10. 

Therefore  any  decimal  may    be    instantly 
written  as  a  fraction;  e.  g.,  0.234  =  2/10  +  3/100  +  4/1000 
=  234/1000. 
First  Method. 

Any  fraction  equals  the  quotient  of  its  numerator 

divided  by  its  denominator.     Consider  the 
2d.  Fractions 

into  fraction,  then,  simply  as  indicating  an  ex- 

ample in  division  of  decimals,  and  proceed 
to  find  the  quotient. 

Thus  for  1/2  we  have :    .  5 

2)1.0    So  1/2  =  0.5. 

For  3/4  we  have     .  75 

4)3.00  So  3/4=0.75. 

For  7/8  we  have:     .875 

8)7.000  So  7/8  =  0.875. 


64          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

Second  Method. 

Apply  the  principle :  The  value  of  a  fraction  is  un- 
altered by  multiplying  both  numerator  and  denominator 
by  the  same  number. 

Thus  7/8-7/(2x2x2) 

=  (7x5x5x5)/(2x5x2x5x2x5) 
=  875/1000-0.875. 

Considering  the  application  of  this  second  method  to 
1/3,  we  see  there  is  no  multiplier  which  will  convert  3 
into  a  power  of  10,  since  10  contains  no  factors  but  2 
and  5.  Ten  does  not  contain  3  as  a  factor,  so  we  cannot 
convert  1/3  into  an  ordinary  decimal.  We  cannot,  as  an 
example  in  division  of  decimals,  divide  1  by  3  without 
remainder.  But  we  can  freely  apply  remainder-division, 
at  any  length.  Thus  333 

3)1. 
.1 

.01 
.001 
The  procedure  is  recurrent,  and  if  continued  the  3 

would  simply  recur. 

.  142857  In  division  by  n,  not  more  than  n—  1 

7)  1 .  different  remainders  can  occur.     But  as 

.3  soon  as  a  preceding  dividend  thus  re- 

2  curs,  the  procedure  begins  to  repeat  it- 

6  self.     Here  then  this  division  by  7  must 

4        begin  to  repeat,  and  the  figures  in  the 

5       quotient  must  begin  to  recur. 

1 

If  the  recurring  cycle  begins  at  once,  immediately 
after  the  decimal  point,  the  decimal  is  called  a  pure  re- 
curring decimal.  As  notation  for  a  pure  recurring  deci- 


RELATION  OF  DECIMALS  TO  FRACTIONS.       65 

mal,  we  write  the  recurring  period,  the  repetend,  dotting 
its  first  and  last  figures  thus 

1/11  =-6$;  1/9  =•!. 

Every  fraction  is  a  product  of  a  decimal  by  a  pure 
recurring  decimal.  Thus 

l/6=(l/2)(l/3)=0.5X-3. 
To  convert  recurring  decimals  into  fractions : 

•  12x     100  =  12-12 

•  12x         1=     -12          Therefore  subtracting, 

•  12x       99  =  12 
.12  =  12/99=4/33 

Rule :  Any  pure  recurring  decimal  equals  the  fraction 
with  the  repeating  period  for  a  numerator,  and  that 
many  nines  for  denominator. 

The  base  of  a  number  system  is  the  number  which 
indicates  how  many  units  are  to  be  taken  together  into  a 

composite  unit,  to  be  named,  and  then  to 
Base.  ,    .  .  ... 

be  used  in  the  count  instead  of  the  units 

composing  it,  this  first  composite  unit  to  be  counted  until, 
upon  reaching  as  many  of  them  as  units  in  the  base,  this 
set  of  composite  units  is  taken  together  to  make  a  com- 
plex unit,  to  be  named,  and  in  turn  to  be  used  in  the 
count,  and  enumerated  until  again  the  basal  number  of 
these  complex  units  be  reached,  which  manifold  is  again 
to  be  made  a  new  unit,  named,  etc. 

Thus  twenty-five,  twain  ten  +  five,  uses  ten  as  base. 
Using  twelve  as  base,  it  would  be  two  dozen  and  one. 
Using  twenty,  it  would  be  a  score  and  five.  In  positional 
notation  for  number,  a  digit  in  the  units'  place  means 
so  many  units,  but  in  the  first  place  to  the  left  of  units' 
place  it  means  so  many  times  the  base,  while  in  the  first 


66  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

place  to  the  right  of  the  units'  place  it  means  so  many 
subtmits  each  of  which  multiplied  by  the  base  gives  the 
unit.  And  so  on,  for  the  second,  etc.,  place  to  the  left 
of  the  units'  column,  and  for  the  second,  etc.,  place  to  the 
right  of  the  units'  column. 

It  is  the  systematic  use  of  a  base  in  connection  with 
the  significant  use  of  position,  which  constitutes  the  for- 
mal perfection  of  our  Hindu  notation  for  number.  The 
actual  base  itself,  ten,  is  a  concession  to  our  fingers. 

The  complete  formula  for  a  number  in  the  Hindu 
positional  notation  is 

dbn.  . .  .  +  db*  +  db3  +  db2  +  db1  +  db()+db-1+db-2+ db~n 

where  juxtaposition  of  the  d  (digit)  and  b  (base)  means 
multiplication.  This  we  condense  to  d.  .  .ddd.ddd.  .  .d, 
where  the  omitted  ^-factor  is  indicated  by  the  position 
of  the  d  with  reference  to  the  units  column,  fixed  by  the 
unital  point  written  to  its  right  in  the  ordered  row. 
Juxtaposition  here  means  addition.  If  no  base  be  speci- 
fied, ten  is  understood. 

Compare  these  subunital  expressions  for  the  funda- 
mental fractions,  to  base  ten,  to  base  twelve,  to  base  two. 

DECIMALLY.      DUODECIMALLY.  DUALLY. 

[IN  THE  DENARY  [DUODENARY  [DYADIC 

SCALE.]  SCALE.]  SCALE.] 

1/2=0-5  1/2  =  0-6  1/2  =  1/10  =0-1 

1/3=  .3  1/3=0-4  1/3  =  1/11  =  -61 

2/3=  -6  2/3=  -8  1/4  =  1/100  =  -01 

1/4=0-25  1/4=0-3  1/6=1/110  =  -OOl 

3/4=  -75  3/4=  .9  1/8  =  1/1000=  -001 

1/5=  .2  1/5  =-249+  1/9  =  1/1001=  -OOOlli 

1/6=0-16  1/6=0-2 

1/8=0-125  1/8=0-16 

3/8=  .375  3/8=  -46 

1/9=  -1  1/9=0-14 


RELATION  OF  DECIMALS  TO  FRACTIONS.       67 

To  express  a  given  number  to  a  new  base,  divide  it 
Change  of  and  the  successive  quotients  by  the  new  base 
base.  until  a  quotient  is  reached  less  than  the  new 

base;  this  quotient  and  the  successive  remainders  will  be 
digits. 
Express  1594  to  base  twelve. 

1  1  -   0     Using  x  for  ten  and  s  for  eleven,  the 
12)132-10     answer  is  sOx. 
12)1594 
Express  sxs  (base  twelve)  to  base  ten. 

1-7       Answer  1715. 
1)15-1 
x)123-5 


Express  98  to  base  two. 

1-1       Answer  1100010. 
2)3-0 
2)6-0 
2)12-0 
2)24-1 
2)49-0 
2)98 
Express  1111   (base  two)  to  base  ten. 

1-101       Answer  15. 
1010)1111 


CHAPTER  XI. 

MEASUREMENT. 

Says  Dr.  E.  W.  Hobson:  "It  is  a  very  significant 
fact  that  the  operation  of  counting,  in  connection  with 
which  numbers,  integral  and  fractional,  have  their  origin, 
is  the  one  and  only  absolutely  exact  operation  of  a  mathe- 
matical character  which  we  are  able  to  undertake  upon 
the  objects  which  we  perceive.  On  the  other  hand,  all 
operations  of  the  nature  of  measurement  which  we  can 
perform  in  connection  with  the  objects  of  perception 
contain  an  essential  element  of  inexactness.  The  theory 
of  exact  measurement  in  the  domain  of  the  ideal  objects 
of  abstract  geometry  is  not  immediately  derivable  from 
intuition." 

Arithmetic  is  a  fundamental  engine  for  our  creative 
construction  of  the  world  in  the  interests  of  our  dom- 
inance over  it.  The  world  so  conceived  bends  to  our  will 
and  purpose  most  completely.  No  rival  construct  now 
exists.  There  is  no  rival  way  of  looking  at  the  world's 
discrete  constituents.  One  of  the  most  far-reaching 
achievements  of  constructive  human  thinking  is  the  arith- 
metization  of  that  world  handed  down  to  us  by  the  think- 
ing of  our  animal  predecessors. 

In  regard  to  an  aggregate  of  things,  why  do  we  care 

„..  .    to  inquire  "how  many"  ?    Why  do  we  count 

Why  count?  .    ,  /       _  ,_.,  J 

an  assemblage  of  things?  Why  not  be  satis- 
fied to  look  upon  it  as  an  animal  would?  How  does  the 
cardinal  number  of  it  help? 


MEASUREMENT.  69 

First  of  all  it  serves  the  various  uses  of  identification. 
Then  the  inexhaustible  wealth  of  properties  individual 
and  conjoined  of  exact  science  is  through  number  assimi- 
lated and  attached  to  the  studied  set,  and  its  numeric 
potential  revealed.  Mathematical  knowledge  is  made  ap- 
plicable and  its  transmission  possible. 

Thus  the  number  is  basal  for  effective  domination  of 
the  world  social  as  well  as  natural, 

Number  arises  from  a  creative  act  whose  aim  and 
purpose  is  to  differentiate  and  dominate  more  perfectly 
than  do  animals  the  perceived  material,  primarily  when 
perceived  as  made  of  individuals.  Not  merely  must  the 
material  be  made  of  individuals,  but  primarily  it  must 
be  made  of  individuals  in  a  way  amenable  to  treatment 
of  this  particular  kind  by  our  finite  powers.  Powers 
which  suffice  to  make  specific  a  clutch  of  eggs,  say  a 
dozen,  may  be  transcended  by  the  stars  in  the  sky. 

Number  is  the  outcome  of  an  aggressive  operation 
of  mind  in  making  and  distinguishing  certain  multiplex 
objects,  certain  manifolds.  We  substitute  for  the  things 
of  nature  the  things  born  of  man's  mind  and  more  obe- 
dient, more  docile.  They,  responsive  to  our  needs,  give 
us  the  result  we  are  after,  while  economizing  our  output 
of  effort,  our  life.  The  number  series,  the  ordered  de- 
numerable  discrete  infinity  is  the  prolific  source  of  arith- 
metic progress.  Who  attempts  to  visualize  90  as  a  group 
of  objects?  It  is  nine  tens.  Then  the  fingers  tell  us 
what  it  is,  no  graphic  group  visualization.  First  comes 
the  creation  of  artificial  individuals  having  numeric  qual- 
ity. The  cardinal  number  of  a  group  is  a  selective  rep- 
resentation of  it  which  takes  or  pictures  only  one  quality 
of  the  group  but  takes  that  all  at  once.  This  selective 
picture  process  only  applies  primarily  to  those  particular 


70  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

artificial  wholes  which  may  be  called  discrete  aggregates. 
But  these  are  of  inestimable  importance  for  human  life. 

The  overwhelming  advantages  of  the  number  picture 
led,  after  centuries,  to  a  human  invention  as  clearly  a 
The  measure  device  of  man  for  himself  as  the  telephone, 
device.  This  was  a  device  for  making  a  primitive 

individual  thinkable  as  a  recognizable  and  recoverable 
artificial  individual  of  the  kind  having  the  numeric  qual- 
ity, having  a  number  picture.  This  is  the  recondite  de- 
vice called  measurement. 

Measurement  is  an  artifice  for  making  a  primitive 
individual  conceivable  as  an  artificial  individual  of  the 
group  kind  with  previously  known  elements,  conven- 
tionally fixed  elements,  and  so  having  a  significant  num- 
ber-picture by  which  knowledge  of  it  may  be  transmitted, 
to  any  one  knowing  the  conventionally  chosen  standard 
unit,  in  terms  of  this  previously  known  standard  unit  and 
an  ascertained  number. 

From  the  number  and  the  standard  unit  for  measure 
the  measured  thing  can  be  approximately  reproduced  and 
so  known  and  recovered.  No  knowledge  of  the  thing 
measured  must  be  requisite  for  knowledge  of  the  stand- 
ard unit  for  the  measurement.  This  standard  unit  of 
measure  must  have  been  familiar  from  previous  direct 
perception.  So  the  picturing  of  an  individual  as  three- 
thirds  of  itself  is  not  measurement. 

All  measurement  is  essentially  inexact.  No  exact 
measurement  is  ever  possible. 

Counting  is  essentially  prior  to  measuring.    The  sav- 

Counting         a&e'  making  the  first  faltering  steps,   fur- 
prior  to          nished  number,  an  indispensable  prerequisite 
for  measurement,  long  ages  before  measure- 
ment was  ever  thought  of.     The  primitive  function  of 


MEASUREMENT.   .  71 

number  was  to  serve  the  purposes  of  identification.  Count- 
ing, consisting  in  associating  with  each  primitive  indi- 
vidual in  an  artificial  individual  a  distinct  primitive  indi- 
vidual in  a  familiar  artificial  individual,  is  thus  itself  essen- 
tially the  identification,  by  a  one-to-one  correspondence,  of 
an  unfamiliar  with  a  familiar  thing.  Thus  primitive  count- 
ing decides  which  of  the  familiar  groups  of  fingers  is  to 
have  its  numeric  quality  attached  to  the  group  counted. 
To  attempt  to  found  the  notion  of  number  upon  measure- 
ment is  a  complete  blunder.  No  measurement  can  be 
made  exact,  while  number  is  perfectly  exact. 

Counting  implies  first  a  known  ordinal  series  or  a 
known  series  of  groups;  secondly  an  unfamiliar  group; 
thirdly  the  identification  of  the  unfamiliar  group  by  its 
one-to-one  correspondence  with  a  familiar  group  of  the 
known  series.  Absolutely  no  idea  of  measurement,  of 
standard  unit  of  measure,  of  value  is  necessarily  involved 
or  indeed  ordinarily  used  in  counting.  We  count  when 
we  wish  to  find  out  whether  the  same  group  of  horses 
has  been  driven  back  at  night  that  was  taken  out  in  the 
morning.  Here  counting  is  a  process  of  identification, 
not  connected  fundamentally  with  any  idea  of  a  standard 
measurement-unit-of-reference,  or  any  idea  of  some  value 
to  be  ascertained.  We  may  say  with  perfect  certainty 
that  there  is  no  implicit  presence  of  the  measurement 
idea  in  primitive  number.  The  number  system  is  not  in 
any  way  based  upon  geometric  congruence  or  measure- 
ment of  any  sort  or  kind. 

The  numerical  measurement  of  an  extensive  quantity 
consists  in  approximately  making  of  it,  by  use  of  a  well- 
known  extensive  quantity  used  as  a  standard  unit,  a  col- 
lection of  approximately  equal,  quantitatively  equal,  quan- 
tities, and  then  counting  these  approximately  equal  quan- 


72  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

tities.  The  single  extensive  quantity  is  said  to  be  numer- 
ically measured  in  terms  of  the  convened  standard  quan- 
titative extensive  unit.  Any  continuous  magnitude  is 
measured  by  discreting  it  into  a  standardized  set  and  a 
negligible  residue,  and  counting  the  standard  units  in 
this  set. 

For  measurement,  assumptions  are  necessary  which 
are  not  needed  for  counting  or  number.  Spatial  measure- 
New  ment  depends  upon  the  assumption  that 
assumptions,  there  is  available  a  standard  body  which 
may  be  transferred  from  place  to  place  without  under- 
going any  other  change.  Therein  lies  not  only  an  as- 
sumption about  the  nature  of  space  but  also  about  the 
nature  of  space-occupying  bodies.  Kindred  assumptions 
are  necessary  for  the  measuring  of  time  and  of  mass. 

Now  in  reality  none  of  these  assumptions  requisite 
for  measurement  are  exactly  fulfilled.  How  fortunate 
then  that  number  involves  no  measurement  idea! 

But  still  other  assumptions  are  made  in  measurement. 
After  this  device  for  making  counting  apply  to  some- 
thing all  in  one  piece  has  marked  off  the  parts  which 
are  to  be  assumed  as  each  equal  to  the  standard,  their 
order  is  unessential  to  their  cardinal  number.  But  it  is 
also  assumed  that  such  pieces  may  be  marked  out  be- 
ginning anywhere,  then  again  anywhere  in  what  remains, 
without  affecting  the  final  remainder  or  the  whole  count. 
Moreover  measurement,  even  the  very  simplest,  must 
face  at  once  incommensurability.  Whatever  you  take  as 
standard  for  length,  neither  it  nor  any  part  of  it  is  exactly 
contained  in  the  diagonal  of  the  square  on  it.  This  is 
proven.  But  the  great  probabilities  are  that  your  stand- 
ard is  not  exactly  contained  in  anything  you  may  wish 


MEASUREMENT.  73 

to  measure.  There  is  a  remainder  large  or  small,  per- 
ceptible or  imperceptible.  Measurement  then  can  only  be 
a  way  of  pretending  that  a  thing  is  a  discrete  aggregate 
of  parts  equal  to  the  standard,  or  an  aliquot  part  of  it. 
We  must  neglect  the  remainder.  If  we  do  it  uncon- 
sciously, so  much  the  worse  for  us. 

No  way  has  been  discovered  of  describing  an  object 
exactly  by  counting  and  words  and  a  standard.  Any 
man  can  count  exactly.  No  man  can  measure  exactly. 

Arithmetic  applies  to  our  representation  of  the  world, 
to  the  constructed  phenomena  the  mind  has  created  to 
help,  to  explain,  its  own  perceptions.  This  representa- 
tion of  things  lends  itself  to  the  application  of  arithmetic. 
Arithmetic  is  a  most  powerful  instrument  for  that  order- 
ing and  simplification  of  perception  which  is  fundamental 
for  dominance  over  so-called  nature. 

Measurement  may  be  analyzed  into  three  primary 
procedures:  1°.  The  conventional  acceptance  or  determi- 
nation of  a  standard  object,  the  unit  of  measure.  2°,  The 
breaking  up  of  the  object  to  be  measured  into  pieces  each 
congruent  to  the  standard  object.  3°.  The  counting  of 
these  pieces. 

The  standard  unit  for  any  particular  sort  of  magni- 
tude might  have  been  any  magnitude  of  the  same  kind. 
Race,  locality,  convenience,  chance,  have  contributed  to 
establish  and  maintain  diverse  units  for  magnitudes  of 
the  same  kind,  some  wholly  bad,  stupid,  indefensible, 
like  the  acre  (160  times  30%  square  yards). 

A  magnitude  is  often  measured  indirectly,  perhaps 
by  substituting  for  it  and  its  standard  unit  two  other 
magnitudes  know  to  have  the  same  quantuplicity  rela- 
tion; thus  an  angle  may  be  measured  indirectly  by  using 


74          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

two  arcs ;  the  thermometer  serves  for  the  indirect  meas- 
urement of  a  temperature  by  use  of  two  volumes ;  a  mass 
is  usually  measured  indirectly  by  use  of  two  weights  at 
the  same  station. 


CHAPTER  XII. 
MENSURATION. 

Never  forget  that  no  exact  measurement  is  ever  pos- 
sible, that  no  theorem  of  arithmetic,  algebra,  or  geometry 
could  ever  be  proved  by  measurement,  that  measure  could 
never  have  been  the  basis  or  foundation  or  origin  of 
number. 

But  the  approximate  measurements  of  life  are  im- 
portant, and  the  best  current  arithmetics  give  great  space 
to  mensuration. 

Geometry  is  an  ideal  construct. 

Of  course  the  point  and  the  straight  are  to  be  assumed 

as  elements,  without  definition.  They  are 
Geometry. 

equally  immeasurable,  the  straight  in  Eu- 
clidean geometry  being  infinite.  What  we  first  measure 
and  the  standard  with  which  we  measure  it  are  both  sects. 
A  sect  is  a  piece  of  a  straight  between  two  points,  the 
end  points  of  the  sect.  The  sides  of  a  triangle  are  sects. 

A  ray  is  one  of  the  parts  into  which  a  straight  is  di- 
vided by  a  point  on  it. 

An  angle  is  the  figure  consisting  of  two  coinitial  rays. 
Their  common  origin  is  its  vertex.  The  rays  are  its 
sides. 

When  two  straights  cross  so  that  the  four  angles 
made  are  congruent,  each  is  called  a  right  angle. 

One  ninetieth  of  a  right  angle  is  a  degree  (1°). 

A  circle  is  a  line  on  a  plane,  equidistant  from  a  point 


76  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

of  the  plane  (the  center).    A  sect  from  center  to  circle 
is  its  radius. 

An  arc  is  a  piece  of  a  circle.  If  less  than  a  semicircle 
it  is  a  minor  arc. 

One  quarter  of  a  circle  is  a  quadrant. 

One  ninetieth  of  a  quadrant  is  called  a  degree  of  arc. 

A  sect  joining  the  end  points  of  an  arc  is  its  chord. 

A  straight  with  one,  and  only  one,  point  in  common 
with  the  circle  is  a  tangent. 

To  measure  a  sect  is  to  find  the  number  L  (its  length) 
Length  of  when  the  sect  is  conceived  as  ~Lu  +  r,  where 
a  sect.  u  js  f^  standard  sect  and  r  a  sect  less  than 

u.    In  science,  u  is  the  centimeter. 

Thus  the  length,  L,  of  the  diagonal  of  a  square  centi- 
meter, true  to  three  places  of  decimals,  is  1 .414. 

Since  there  are  different  standard  sects  in  use,  it  is 
customary  to  name  u  with  the  L.  Here  1 .414  cm. 

Knowing  the  length  of  a  sect,  from  our  knowledge 
of  the  number  and  the  standard  sect  it  multiplies  we  get 
knowledge  of  the  measured  sect,  and  can  always  approxi- 
mately construct  it. 

We  assume  that  with  every  arc  is  connected  one,  and 
only  one,  sect  not  less  than  the  chord,  and  if  the  arc 
Length  of  De  minor,  not  greater  than  the  sum  of  the 
the  circle.  sects  on  the  tangents  from  the  extremities 
of  the  arc  to  their  intersection,  and  such  that  if  the  arc 
be  cut  into  two  arcs,  this  sect  is  the  sum  of  their  sects. 
The  length  of  this  sect  we  call  the  length  of  the  arc. 

If  r  be  the  length  of  its  radius,  the  length  of  the 
semicircle  is  nr. 

Archimedes  expressed  IT  approximately  as  3  +  1/7. 

True  to  two  places  of  decimals,  7r  =  3.14  or  3.1416 
true  to  four  places. 


MENSURATION. 

The  approximation  7r  =  3  +  l/7  is  true  to  three  signifi- 
cant figures.  But  since  TT  =  3. 1416  =  3  +  1/7-  1/800,  a 
second  approximation,  true  to  five  significant  figures,  can 
be  obtained  by  a  correction  of  the  first. 

Again  ir  =  3. 1416=  (3  +  1/7)  (1-. 0004),  which  gives 
the  advantage  that  in  a  product  of  factors  including  ir, 
the  value  3  +  1/7  can  be  used  and  the  product  corrected 
by  subtracting  four  ten-thousandths  of  itself. 

The  circle  with  the  standard  sect  for  radius  is  called 
the  unit  circle.  The  length  of  the  arc  of  unit  circle  inter- 
cepted by  an  angle  with  vertex  at  center  is  called  the 
size  of  the  angle. 

The  angle  whose  size  is  1,  the  length  of  the  standard 
sect,  is  called  a  radian. 

A  radian  intercepts  on  any  circle  an  arc  whose  length 
is  the  length  of  that  circle's  radius. 

The  number  of  radians  in  an  angle  at  the  center 
intercepting  an  arc  of  length  L  on  circle  of  radius  length 
r,  is  L/r.  180°=7r/>. 

An  arc  with  the  radii  to  its  endpoints  is  called  a 
sector. 

The  area  of  a  triangle  is  half  the  product  of  the 

length  of  either  of  its  sides  (the  base)  by 

the   length   of  the  corresponding  altitude, 

the  perpendicular  upon  the  straight  of  that  side  from  the 

opposite  vertex. 

A  figure  which  can  be  cut  into  triangles  is  a  polygon, 
whose  area  is  the  sum  of  theirs.  Its  perimeter  is  the  sum 
of  its  sides. 

Area  of  Circle.  In  area,  an  inscribed  regular  polygon 
(one  whose  sides  are  equal  chords)  of  2n  sides  equals 
a  triangle  with  altitude  the  circle's  radius  r  and  base  the 
perimeter  of  an  inscribed  regular  polygon  of  n  sides. 


78          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

A  circumscribed  regular  polygon  (one  with  sides  on 
tangents)  of  n  sides  equals  a  triangle  with  altitude  r  and 
base  the  polygon's  perimeter. 

There  is  one,  and  only  one,  triangle  intermediate  be- 
tween the  series  of  inscribed  regular  polygons  and  the 
series  of  circumscribed  regular  polygons,  namely  that 
with  altitude  r  and  base  equal  in  length  to  the  circle. 
This  triangle's  area,  rc/2  =  r27r,  is  the  area  of  the  circle, 
r2*. 

From  analogous  considerations,  the  area  of  a  sector 
is  the  product  of  the  length  of  its  arc  by  the  length  of 
half  its  radius. 

A  tetrahedron  is  the  figure  constituted  by  four  non- 

coplanar  points,  their  sects  and  triangles. 

Volume.  »«_    *  •  -L  n  j  v  -j. 

The  four  points  are  called  its  summits, 

the  six  sects  its  edges,  the  four  triangles  its  faces. 

Every  summit  is  said  to  be  opposite  to  the  face  made 
by  the  other  three;  every  edge  opposite  to  that  made  by 
the  two  remaining  summits. 

A  polyhedron  is  the  figure  formed  by  n  plane  polygons 
such  that  each  side  is  common  to  two.  The  polygons  are 
called  its  faces;  their  sects  its  edges;  their  vertices  its 
summits. 

One-third  the  product  of  the  area  of  a  face  by  the 
length  of  the  perpendicular  to  it  from  the  opposite  vertex 
is  the  volume  of  the  tetrahedron. 

The  volume  of  a  polyhedron  is  the  sum  of  the  vol- 
umes of  any  set  of  tetrahedra  into  which  it  is  cut. 

A  prismatoid  is  a  polyhedron  with  no  summits  other 
than  the  vertices  of  two  parallel  faces. 

The  altitude  of  a  prismatoid  is  the  perpendicular  from 
top  to  base. 


MENSURATION.  79 

A  number  of  different  prismatoids  thus  have  the  same 
base,  top,  and  altitude. 

If  both  base  and  top  of  a  prismatoid  are  sects,  it  is 
a  tetrahedron. 

A  section  or  cross-cut  of  a  prismatoid  is  the  polygon 
determined  by  a  plane  perpendicular  to  the  altitude. 

To  find  the  volume  of  any  prismatoid.  Rule :  Multi- 
ply one-fourth  its  altitude  by  the  sum  of  the  base  and 
three  times  the  cut,  at  two-thirds  the  altitude  from  the 
base. 

Halsted's  Formula:  V=  (o/4)(B  +  3C). 

All  the  solids  of  ordinary  mensuration,  and  very  many 
others  heretofore  treated  only  by  the  higher  mathematics, 
are  nothing  but  prismatoids  or  covered  by  Halsted's  For- 
mula. 

A  pyramid  is  a  prismatoid  with  a  point  as  top.  Hence 
its  volume  is  aB/3. 

A  circular  cone  is  a  pyramid  with  circular  base. 

A  prism  is  a  prismatoid  with  all  lateral  faces  paral- 
lelograms. 

Hence  the  volume  of  any  prism  =aB. 

A  circular  cylinder  is  a  prism  with  circular  base. 

A  right  prism  is  one  whose  lateral  edges  are  per- 
pendicular to  its  base. 

A  parallelepiped  is  a  prism  whose  base  and  top  are 
parallelograms. 

A  cuboid  is  a  parallelepiped  whose  six  faces  are  rect- 
angles. 

A  cube  is  a  cuboid  whose  six  faces  are  squares. 

Hence  the  volume  of  any  cuboid  is  the  product  of 
its  length,  breadth  and  thickness. 

The  cube  whose  edge  is  the  standard  sect  has  for 
volume  1. 


80          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

Therefore  the  volume  of  any  polyhedron  tells  how 
oft  it  contains  the  cube  on  the  standard  sect,  called  the 
unit  cube. 

Such  units,  like  the  unit  square,  though  traditional, 
are  unnecessary. 

A  sphere  is  a  surface  equidistant  from  a  point  (the 
center). 

A  sect  from  the  center  to  sphere  is  its  radius. 

A  spherical  segment  is  the  piece  of  a  sphere  between 
two  parallel  planes. 

If  a  sphere  be  tangent  to  the  parallel  planes  containing 
opposite  edges  of  a  tetrahedron,  and  sections  made  in  the 
sphere  and  tetrahedron  by  one  plane  parallel  to  these  are 
of  equal  area,  so  are  sections  made  by  any  parallel  plane. 
Hence  the  volume  of  a  sphere  is  given  by  Halsted's  For- 
mula. 

V=  (a/4)  (B  +  3C)  =  (3/4)aC. 

But  a  =  2r  and  C=  (2/3)nr(4/3)r. 

So  Vol.  sphere  =  (4/3)  Trr3. 

Hence  also  the  volume  of  a  spherical  segment  is  given 
by  Halsted's  Formula. 

Area  of  sphere  =  4?rr2. 

The  area  of  a  sphere  is  quadruple  the  area  of  its 
great  circle. 

As  examples  of  solids  which  might  now  be  introduced 
into  elementary  arithmetic,  since  they  are  covered  by  Hal- 
sted's Formula,-  may  be  mentioned :  oblate  spheroid,  pro- 
late spheroid,  ellipsoid,  paraboloid  of  revolution,  hyper- 
boloid  of  revolution,  elliptic  hyperboloid,  and  their  seg- 
ments or  frustums  made  by  planes  perpendicular  to  their 
axes,  all  solids  uniformly  twisted,  like  the  squarethreaded 
screw,  etc. 


CHAPTER  XIII. 
ORDER. 

In  the  counting  of  a  primitive  group,  any  element  is 
considered  equivalent  to  any  other.  But  in  the  use  even 
of  the  primitive  counting  apparatus,  the  fingers,  appeared 
another  and  extraordinarily  important  character,  order. 

If  always  when  any  two  elements  a,  &  of  a  set  are 
taken,  a  definite  criterion  fixes  one  or  other  of  two  alter- 
native relations,  symbolized  by  a  generalized  use  of  > 
and  <,  such  that  if  a  <  b  then  b>  a,  while  if  a  >  b  then 
b  <  a,  and  such  that  if  a  <  b  and  b  <  c,  then  a  <  c,  we 
say  the  criterion  arranges  the  set  in  order.  So  arranged, 
it  becomes  an  ordered  set. 

The  savage  in  counting  systematically  begins  his  count 
with  the  little  finger  of  the  left  hand,  thence  proceeding 
toward  the  thumb,  which  is  fifth  in  the  count.  When 
number-words  or  number-symbols  come  to  serve  as  ex- 
tended counting  apparatus,  order  is  a  salient  character- 
istic. Each  is  associated  with  a  definite  next  succeeding 
number.  The  set  possesses  intrinsic  order. 

By  one-to-one  adjunction  of  these  numerals  the  in- 
dividuals of  a  collection  are  given  a  factitious  order,  the 
familiar  order  of  the  number-set. 

When  the  order  is  emphasized  the  number-names  are 
modified,  becoming  first,  second,  third,  fourth,  etc.,  and 
are  called  ordinal  numbers  or  ordinals,  but  this  designa- 
tion is  now  applied  also  to  the  ordinary  forms,  one,  two, 


82          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

three,  etc.,  when  order  is  made  their  fundamental  char- 
acteristic. 

If  we  can  so  correlate  each  element  of  the  set  A  with 
a  definite  element  of  the  set  B  that  two  different  elements 

of  A  are  never  correlated  with  the  same 
Depiction.  .  _.  ..    .    .  .  , 

element  of  B,  the  element  of  A  is  consid- 
ered as  depicted  or  pictured  or  imaged  by  the  correlated 
element  of  B,  its  picture  or  image. 

Such  a  correlation  we  call  a  depiction  of  the  set  A 
upon  the  set  B.  The  elements  of  A  are  called  the  originals. 

An  assemblage  contained  entirely  in  another  is  called 
a  component  of  the  latter. 

A  proper  component  or  proper  part  of  an  assemblage 
is  an  aggregate  made  by  omitting  some  element  of  the 
assemblage. 

An  assemblage  is  called  infinite  if  it  can  be  depicted 

_      .  upon  some  proper  part  of  itself,  or  distinctly 

Infinite.  -11  <•         <  1 

imaged,  element  for  element,  by  a  constit- 
uent portion,  a  proper  component  of  itself.  Otherwise 
it  is  finite. 

Stand  between  two  mirrors  and  face  one  of  them. 
Your  image  in  the  one  faced  will  be  repeated  by  the 
other.  If  this  replica  could  be  separately  reflected  in  the 
first,  this  reflection  imaged  by  itself  in  the  second,  this 
image  pictured  as  distinct  in  the  first,  this  in  turn  depicted 
in  the  second,  and  so  on  forever,  this  set  would  be  in- 
finite, for  it  is  depicted  upon  the  proper  part  of  it  made 
by  omitting  you.  It  is  ordered.  You  may  be  called  1, 
your  image  2,  its  image  3,  and  so  on. 

A  relation  has  what  mathematicians  call  sense,  if, 
when  A  has  it  to  B,  then  B  has  to  A  a  relation 
different,  but  only  in  being  correlatively  op- 
posite. Thus  "greater  than"  is  a  sensed  relation.  "Greater 


ORDER.  83 

than"  and  "less  than"  are  different  relations,  but  differ 
only  in  sense. 

Any  number  of  numbers,  all  individually  given,  form 
a  finite  set.  If  numbers  be  potentially  given  through  a 
given  operand  and  a  given  operation,  law,  of  successive 
eduction,  they  are  still  said  to  form  a  set.  If  the  law 
educes  the  numbers  one  by  one  in  definite  succession, 
they  have  an  order,  taking  on  the  order  inherent  in  time 
or  in  logical  or  causal  succession. 

A  set  in  order  is  a  series. 

Intrinsic  order  depends  fundamentally  upon  relations 
Analysis  of  having  sense,  and,  for  three  terms,  upon  a 
order.  relation  and  its  opposite  in  sense  attaching 

to  a  given  term. 

The  unsymmetrical  sensed  relation  which  determines 
the  fixed  order  of  sequence  may  be  thought  of  as  a  logic- 
relation,  that  an  element  shall  involve  a  logically  sequen- 
tial element  creatively  or  as  representative.  An  individual 
or  element  1  has  its  shadow  2,  which  in  turn  has  its 
shadow  3,  and  so  on. 

Linear  order  is  established  by  an  unsymmetrical  re- 
lation for  one  sense  of  which  we  may  use  the  word  "pre- 
cede," for  the  opposite  sense  "follow." 

The  ordering  relation  may  be  envisaged  as  an  opera- 
tion, a  transformation,  which  performed  upon  a  preced- 
ing gives  the  one  next  succeeding  it;  turns  1  into  2,  and 
2  into  3,  and  so  on. 

If  we  have  applicable  to  a  given  individual  an  opera- 
tion which  turns  it  into  a  new  individual  to  which  in 
turn  the  operation  is  applicable  with  like  result,  and  so 
on  without  cease,  we  have  a  recurrent  operation  which 
recreates  the  condition  for  its  ongoing.  If  in  such  a 
set  we  have  one  and  only  one  term  not  so  created  from 


84  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

any  other,  a  first  term,  and  if  every  term  is  different 
from  all  others,  we  have  a  commencing  but  unending 
ordered  series.  The  number  series,  1,  2,  3,  and  so  on, 
may  be  thought  of  as  the  outcome  of  a  recurrent  opera- 
tion, that  of  the  ever  repeated  adjunction  of  one  more 
unit.  It  is  a  system  such  that  for  every  element  of  it 
there  is  always  one  and  only  one  next  following.  This 
successor  may  be  thought  of  as  the  depiction  of  its  pred- 
ecessor. Every  element  is  different  from  all  others. 
Every  element  is  imaged.  There  is  an  element  which 
though  imaged  is  itself  no  image. 

Thus  the  series  is  depicted  without  diminution  upon 
a  proper  part  of  itself;  is  infinite,  and  by  constitution 
endless.  It  has  a  first  element,  but  no  element  following 
all  others,  no  "last"  element. 

Any  set  which  can  be  brought  into  one-to-one  cor- 
respondence with  some  or  all  of  the  natural  numbers  is 
said  to  be  countable,  and  if  not  finite,  is  called  countably 
infinite. 

An  order  of  a  set  is  constituted  by  a  relation  between 

the  elements  of  the  set.    The  same  set  may 
Ordered  set.    ,  .  ,.„ 

have  at  the  same  time  many  different  orders. 

The  particular  order  is  defined  by  the  particular  serial  or 
arranging  relation. 

A  set  of  elements  is  said  to  be  in  simple  order  if  it 
has  two  characteristics: 

1°.  Every  two  distinct  arbitrarily  selected  elements, 
A  and  B,  are  always  connected  by  the  same  unsym- 
metrical  relation,  in  which  relation  we  know  what  role 
one  plays,  so  that  always  one,  and  only  one,  say  A,  comes 
before  B,  is  source  of  B,  precedes  B,  is  less  than  B; 
while  B  comes  after  A,  is  derived  from  A,  follows  A, 
is  greater  than  A. 


ORDER. 


85 


2°.  Of  three  elements  ABC,  if  A  precedes  B,  and  B 
precedes  C,  then  A  precedes  C. 

So  an  arranging  relation  implies  diversity  of  the  ele- 
ments, is  transitive,  and  connects  any  two  different  ele- 
ments related  by  it  to  a  third.  Thus  the  moments  of 
time  between  twelve  and  one  o'clock,  and  the  points  on 
the  sect  AB  as  passed  in  going  from  A  to  B  are  simply 
ordered  sets. 

Two  ordered  sets  A,  B  are  called  similar  when  a  one- 
to-one  correspondence  can  be  established  between  their 
elements  such  that  if  a<a'  in  A  then  their  correlates 
b  <  b'.  Similar  ordered  sets  are  said  to  have  the  same 
order-type. 

An  arranged  finite  set  of,  say,  n  elements  can  be 
Finite  ordinal  brought  into  one-to-one  correspondence  with 
types.  tne  first  n  integers. 

Such  an  ordered  set  has  a  first  and  a  last  element; 
so  has  each  ordered  component. 

Inversely  an  ordered  set  with  a  first  and  last  element, 
whose  every  component  has  a  first  and  a  last  element,  is 
finite.  For  let  a\  be  the  first  element.  The  remaining 
elements  form  an  ordered  component;  let  02  be  the  first 
of  these  elements.  In  the  same  way  determine  as.  We 
must  thus  reach  the  last,  else  were  there  an  ordered  com- 
ponent without  last  element,  contrary  to  hypothesis.  These 
then  are  the  characteristics  of  the  finite  ordinal  types. 

Any  set  equivalent  to  the  natural  number  series  (the 
Numb  r  natural  scale)  is  called  countably  infinite. 
series,  type  The  characteristic  property  of  a  count- 
ably  infinite  set,  when  arranged  in  count- 
able order,  is  that  we  know  of  any  element  a  whether, 
or  no,  it  corresponds  to  a  smaller  integer  than  does  the 
element  b.  Should  a  and  b  correspond  to  the  same  in- 


86          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

teger  they  would  be  identical.  Thus  when  arranged  in 
countable  order,  the  order  of  any  countably  infinite  set 
is  that  of  the  natural  numbers.  The  defining  character- 
istics of  this  ordered  set  are  that  it,  as  well  as  each  of 
its  ordered  components,  has  a  first  element,  and  that 
every  element,  except  the  first,  has  another  immediately 
preceding  it;  while  each  element  has  one  next  following, 
and  consequently,  there  is  no  last  element. 

Any  simply  ordered  set  between  any  pair  of  whose 
elements  there  is  always  another  element  is  said  to  be  in 
close  order. 

A  simply-ordered  set  is  said  to  be  "well-ordered" 
Well-ordered  ^  the  set  itself,  as  well  as  every  one  of  its 
sets*  components,  has  a  first  element. 

In  a  well-ordered  set  its  elements  so  follow  one  an-v 
other  according  to  a  given  law  that  every  element  is  im- 
mediately followed  by  a  completely  determined  element, 
if  by  any.  As  typical  of  well-ordered  sets  we  may  take 
first  the  finite  sets  of  the  ordinal  numbers:  1st;  1st,  2d; 
1st,  2d,  3d;  and  so  on. 

As  typical  of  the  first  transfinite  well-ordered  set  we 
may  take  the  set  of  all  the  ordinal  numbers,  the  ascend- 
ing order  of  the  natural  numbers. 

The  thousandth  even  number  is  immediately  followed 
by  the  number  2001. 

But  if  a  point  B  is  taken  on  a  sect  AC,  there  is  no 
next  consecutive  point  to  B  determinable. 

The  way  in  which  an  iterative  operation  develops 
from  an  individual  operand  not  only  infinity  but  endless 
variety  unthought  of  and  so  waiting  to  be  thought  of, 
lights  up  the  fact  that  mathematics  though  deductive  is 
not  troubled  with  the  syllogism's  tautology  but  offers 
ever  green  fields  and  pastures  new.  Thus  in  the  number 


ORDER.  87 

series  is  the  series  of  even  numbers,  in  this  the  set  of  even 
even  numbers,  4,  8,  12,  16,  20,  etc.,  each  a  system  in 
which  every  element  of  every  preceding  system  of  this 
series  of  systems  can  have  its  own  uniquely  determined 
picture,  the  first  term  depicting  any  first  term,  the  second 
any  second,  etc. 


CHAPTER  XIV. 
.     ORDINAL  NUMBER. 

Numbers  are  ordinal  as  individuals  in  a  well-ordered 
Ordinal  set  or  series,  and  used  ordinally  when  taken 

number.  to  gjve  to  any  one  object  its  position  in  an 

arrangement  and  thus  to  individually  identify  and  place 
it  in  a  series. 

The  ordinal  process  has  also  as  outcome  knowledge 
of  the  cardinal ;  when  we  have  in  order  ticketed  the  ninth, 
we  have  ticketed  nine.  Thus  the  last  ordinal  used  tells 
the  result  of  the  count,  being  given  a  cardinal  meaning 
to  denote  the  particular  plurality  of  the  set  now  ticketed. 

The  assignment  of  order  to  a  collection  and  ascer- 
tainment of  place  in  the  series  made  by  this  putting  in 
Children's  order  is  shown  by  that  use  of  count  which 
counting.  occurs  in  children's  games,  in  their  counting 
out  or  counting  to  fix  who  shall  be  it.  This  counting  is 
the  use  of  a  set  of  words  not  ever  investigated  as  to 
multiplicity,  but  characterized  by  order.  Such  is  the 
actually-used  set:  ana,  mana,  mona,  mike;  bahsa,  lona, 
bona,  strike;  hare,  ware,  frounce,  nack;  halico,  baliko, 
we,  wo,  wy,  wak.  Applied  to  an  assemblage,  it  gives 
order  to  the  assemblage  until  exhausted,  and  the  last 
one  of  the  ordered  but  unnumbered  group  is  out  or  else 
it.  How  many  individual  wprds  the  ordering  group  con- 
tains is  never  once  thought  of.  There  is  successive  enu- 
meration without  simultaneous  apprehension. 


ORDINAL  NUMBER.  89 

Every  element  has  an  ordinal  significance.  No  ele- 
ment has  any  cardinal  significance. 

E  nee.  me  nee,  my  nee,  mo; 
Crack  ah,  fee  nee,  f y  nee,  f o ; 
Amo  neu  ger,  po  po  tu  ger ; 
Rick  stick,  jan  jo. 

Such  a  group  but  indefinitely  extensible,  having  a  first 
but  no  last  term,  is  the  ordinal  number  series. 

But  in  our  ordinary  system  of  numeral  words,  with 
fixed  and  rote-learned  order,  each  word  is  used  to  con- 
Uses  of  vey  also  an  exact  notion  of  the  multiplicity 
ordinals.  of  individuals  in  the  group  whose  tagging 
has  used  up  that  and  all  preceding  numerals.  Thus  each 
one  characterizes  a  specific  group,  and  so  has  a  cardinal 
content. 

Yet  it  is  upon  the  ordered  system  itself  that  we  chiefly 
rely  to  get  a  working  hold  of  the  number  when  beyond 
the  point  where  we  try  to  have  any  complete  appreciation, 
as  simultaneous,  of  the  collection  of  natural  units  in- 
volved. Thus  it  is  to  the  ordinal  system  that  we  look  for 
succor  and  aid  in  getting  grasp  and  understanding  par- 
ticularly of  numbers  too  great  for  their  component  indi- 
vidual units  to  be  at  once  and  together  separately  pic- 
turable.  Thus  the  ideas  we  get  of  large  numbers  come 
not  from  any  attempt  to  realize  the  multiplicity  of  the 
discrete  manifold,  but  rather  from  place  in  the  number- 
set. 

Number  in  its  genesis  is  independent  of  quantity,  and 
number-science  consists  chiefly,  perhaps  essentially,  in  re- 
lations of  one  number  in  the  number  series  to  another 
and  to  the  series. 

That  a  concept  is  dependent  for  its  existence  upon  a 
word  or  language-symbol  is  a  blunder.  The  savage  has 


90          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

number-concepts  beyond  words.  On  the  other  hand,  the 
modern  child  gets  the  words  of  the  ordinal  series  before 
the  cardinal  concepts  we  attach  to  them.  If  a  little  child 
says,  "Yes,  I  can  count  a  hundred,"  it  simply  means  it 
can  repeat  the  series  of  number-names  in  order.  Its  slips 
would  be  skips  or  repetitions.  The  ordinal  idea  has  been 
formed.  It  is  used  by  the  child  who  recognizes  its  errors 
in  this  ordinal  counting.  The  ordinal  idea  has  been  made, 
has  been  embodied  perhaps  in  rythmical  movements.  The 
child's  rudimentary  counting  set  is  a  sing-song  ditty.  The 
number  series  when  learned  is  perhaps  chanted.  Just  so 
there  is  a  pleasurable  swing  in  the  count  by  fives. 

The  use  of  the  terms  of  the  number  series  as  instru- 
ments for  individual  identification  appears  in  the  primi- 
tive child's  game.  Before  making  or  using  number,  chil- 
dren delight  in  making  series.  Succession  is  one  of  the 
earliest  made  thoughts. 

We  think  in  substituted  symbols.  It  is  folly  to  at- 
tempt to  hold  back  the  child  in  this  substitution.  The 
abstractest  number  becomes  a  thing,  an  objective  reality. 

Number  has  not  originated  in  comparison  of  quantity 
nor  in  quantity  at  all.  Number  and  quantity  are  wholly 
independent  categories,  and  the  application  of  number  to 
quantity,  as  it  occurs  in  measurement,  has  no  deeper  mo- 
tive than  one  of  convenience. 

It  has  often  been  stressed,  that  children  knowing  the 
number-names,  if  asked  to  count  objects,  pay  out  the 
series  far  faster  than  the  objects ;  the  names  far  outstrip 
the  things  they  should  mate. 

The  so-called  passion  of  children  for  counting  is  a 
delight  in  ordinal  tagging,  in  ordinal  depiction  with 
names,  with  no  attempt  to  carry  the  luggage  of  cardinals. 

The  "which  one"  is  often  more  primitive  and  more 


ORDINAL  NUMBER.  91 

important  than  the  "how  many."  The  hour  of  the  day 
is  an  ordinal  in  an  ordered  set.  Its  interest  for  us  is 
wholly  ordinal.  It  identifies  one  element  in  an  ordered 
set.  The  strike  of  the  clock  is  a  word.  The  striking 
clock  has  a  vocabulary  of  12  words.  These  words  are 
distinguished  by  the  cardinal  number  of  their  syllables. 
But  even  when  recognized  by  the  cardinal  number  of 
syllables  in  its  clock-spoken  name,  the  hour  is  in  essence 
an  ordinal. 

So  the  number  series  as  a  word-song  may  well  in  our 
children  precede  any  application  to  objects.  Objects  are 
easily  over-estimated  by  those  who  have  never  come  to  the 
higher  consciousness  that  objects  are  mind-made,  that 
every  perception  must  partake  of  the  subjective. 

Children  often  apply  the  number-names  to  natural 
individuals  as  animals  might,  that  is  without  making 
any  artificial  or  man-made  individual,  and  so  without  any 
cardinal  number.  Each  name  depicts  a  natural  individ- 
ual, but  not  as  component  of  a  unity  composed  of  units. 
What  passes  for  knowledge  of  number  among  animals 
is  only  recognition  of  an  individual  or  an  individual 
form. 

Serial  depiction  under  the  form  of  tallying  or  beats 
or  strokes  may  precede  all  thought  of  cardinal  number. 
Nine  out  of  ten  children  learn  number  names  merely  as 
words,  not  from  objects  or  groups. 

The  typical  case  is  given  of  the  girl  who  could  "count" 
100  long  before  she  could  recognize  a  group  of  seven 
objects. 

The  names  of  the  natural  numbers  are  an  unending 
child's  ditty,  primarily  ordinal,  but  a  ditty  to  whose  terms 
cardinal  meanings  have  also  been  attached.  Ordinally 
the  number  name  "one"  is  simply  the  initial  term  of  this 


92  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

series;  any  number  name  is  simply  a  term  of  this  series. 
The  ordinal  property  it  designates  is  the  positional  prop- 
erty of  an  element  in  a  well-ordered  set,  the  place  of  an 
individual  in  a  series. 

The  natural  scale  is  the  standard  for  civilized  count- 
ing. Its  symbols  in  sequence  are  mated  with  the  elements 
of  an  aggregate  and  the  last  symbol  used  gives  the  out- 
come of  the  count,  tells  the  cardinal  number  of  the 
counted  aggregate.  The  cardinal,  n,  of  a  set  is  that  at- 
tribute by  which  when  the  set's  elements  are  coupled  with 
ordinals,  the  ordinal  n  and  all  ordinals  preceding  n  are 
used. 

The  very  first  step  in  the  teaching  of  arithmetic  should 
be  the  child's  chanting  of  the  number  names  in  order. 
Then  the  first  application  should  be  ordinal.  Use  the 
numbers  as  specific  tags,  conveying  at  first  only  order 
and  individual  identification.  Afterward  connect  with 
each  group,  as  its  name,  the  last  numeral  it  uses,  which 
thus  takes  on  a  cardinal  significance. 

Modern  civilization  has  brought  out  a  use  of  numbers 
neither  ordinal  nor  cardinal.  It  is  their  employment  as 
Nominal  mere  proper  nouns.  His  number  is  the  con- 
number,  vict's  name.  This  use  of  what  may  be  called 
nominal  number  has  reached  its  highest  social  develop- 
ment in  the  telephone.  Since  the  size  of  the  number  and 
its  place  in  the  number  series  are  here  alike  irrelevant, 
the  whole  stress  falls  upon  its  recognition  as  a  unique 
name  made  by  the  juxtaposition  in  linear  order  of  ten 
simple  symbols,  the  nine  digits  and  the  zero.  And  these 
symbols  must  be  orally  conveyed  to  a  girl  whose  vocabu- 
lary is  so  meager  it  does  not  contain  the  word  triple.  So 
333  is  read  three,  three,  three.  But  the  profoundest 
development  is  that  zero  has  dropped  everything  but  its 


ORDINAL  NUMBER.  93 

adventitious  Italian  ending  o,  and  so  evolved  a  new  name, 
oh!  Thus  The  Saturday  Evening  Post,  Aug.  5,  1911, 
p.  11,  has  "Six-oh-nine-two  Nassau";  for  the  telephone 
rejoices  also  in  a  family  name.  Thus  "The  Thousand 
and  One  Nights,"  as  a  telephone  name  reads :  One- 
oh-oh-one  Nights.  But  surely  in  these  proper  names 
the  family  should  come  first  as  in  Magyar,  Bolyai  John, 
and  we  should  have  Greeley  oh-oh-oh.  Since  both  in- 
trinsic and  local  value  have  vanished,  there  are  111  more 
nominal  than  cardinal  numbers  before  100.  Among  nom- 
inal numbers,  the  additional  class  corresponding  to  no 
new  cardinals  may  be  called  roundheads,  e.  g.,  00,  01,  02, 
etc.,  000,  030,  099. 


CHAPTER  XV. 
THE  PSYCHOLOGY  OF  READING  A  NUMBER. 

Our  marvelous  positional  notation  for  number  is  built 
of  three  elements,  digit,  base,  column.  The  base  it  is 
which  interprets  the  column.  With  base  ten,  100  means 
a  ten  of  tens.  With  base  two,  100  means  two  twos.  With 
base  twelve,  100  means  a  dozen  dozen. 

The  Romans  had  a  base,  or  rather  two  bases,  but 
neither  digits  nor  columns.  Their  V  is  a  trace  of  the 
more  primitive  base  five,  seen  also  in  the  Greek  7re^7ra£e0, 
to  finger  fit  by  fives,  to  count.  This,  combining  with 
the  more  final  base  ten,  X,  explains  their  having  a  sepa- 
rate symbol,  L,  for  fifty,  and  D  for  five  hundred. 

Their  ten  of  tens  has  its  unitary  symbol,  C,  and  their 
ten  of  hundreds  is  M,  a  thousand. 

Each  basal  number  is  a  new  unit,  an  atom,  a  monad, 
a  neomon,  squeezing  into  an  individual  the  components, 
making  thus  one  ball  to  be  further  played  with. 

Our  present  basal  number-word,  hundred,  is  properly 
a  collective  noun,  a  hundred,  literally  a  tenth  count 
or  tale;  for  its  red  is  the  root  in  German  Rede,  talk, 
our  rate,  reckon,  and  its  hund  is  the  Old  English  word, 
cognate  with  Latin  centum,  Greek  efcarov,  to  be  found 
in  Bosworth's  Anglo-Saxon  Dictionary,  but  seldom  used 
after  A.  D.  1200. 

The  Century  Dictionary,  to  which  I  may  be  forgiven 
for  being  attached,  says  hund  is  from  the  root  of  ten, 


THE   PSYCHOLOGY   OF   READING   A   NUMBER.  95 

and  this  leads  it  far,  into  the  postulating  of  an  assumed 
type  kanta  which  it  gives  as  a  reduced  form  of  an  equally 
hypothetical  dakanta  for  an  assumed  original  dakan- 
dakan-ta,  "ten-ten-th,"  from  assumed  dakan,  on  the  anal- 
ogy of  the  Gothic  taihun-taihund,  taihun-tehund,  a  hun- 
dred, of  which  it  regards  hund  as  an  abbreviation  or  re- 
duced form.  The  same  original  elements,  it  says,  with- 
out the  suffix  d  -  th,  appear  in  Old  High  German  zehanzo 
=  Anglo-Saxon  t eon-tig,  ten-ty  =  ten-ten. 

The  element  hund  occurring  in  the  Anglo-Saxon 
hund-seofontig,  seventy,  etc.,  hund-endlefontig,  eleventy, 
hund-twelftig,  twelfty,  it  gives  as  representing  "ten"  or 
"tenth,"  and  these  words  as  developed  by  cumulation 
(hund  and  tig  being  ultimately  from  the  same  root,  that 
of  "ten")  from  the  theoretically  assumed  hund-seofon, 
"tenth  seven,"  etc.  Murray  is  not  well  persuaded  of  all 
this,  and  says  there  is  no  satisfactory  explanation  of  the 
use  of  hund  in  these  Anglo-Saxon  words. 

However  that  may  be,  just  as,  in  Latin,  de-cem  gives 
centum,  so  t-enth  gives  hund,  in  each  case  the  dental,  or 
better,  lingual,  dropping  away.  Moreover,  with  us  this 
enth  or  hund,  with  Saxon  dogged  persistence,  reappears 
in  thous-and,  as  shown  by  the  Icelandic  thusund,  thils- 
hund,  thushundrad,  though  Latin  here  takes  a  new  start 
with  mille,  the  Sanskrit  root  mil,  to  unite,  to  combine, 
seen  also  in  miles,  a  soldier,  and  militia.  Perhaps  our 
prefix  thous,  Icelandic  thus,  is  Teutonic  thu,  Aryan  tu, 
to  swell,  seen  in  tumor. 

So  our  "a  hundred"  is  an  abbreviation  for  the  phrase 
"a  tenth  reckoning  [of  decads]." 

This  is  consonant  with  the  fact  that  in  Old  Norse 
the  word  hundrath,  "hundred,"  "tenthtale,"  originally 
meant  120;  it  was  a  tenthtale  not  of  tens  but  of  dozens, 


96          FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

the  rival  base  twelve,  against  which  the  bestial  base  ten, 
an  Old-Man-of-the-Sea  saddled  upon  us  by  our  pre- 
human simian  ancestors,  has  been  continuously  fighting 
down  to  this  very  day.  And  even  in  modern  English, 
remnants  of  this  older  usage  remain.  The  Glasgozv 
Herald  of  September  13,  1886,  says:  "A  mease  [of 
herring] ...  .is  five  hundreds  of  120  each." 

Chambers  Cyclopaedia  says:  "Deal  boards  are  six 
score  to  the  hundred." 

This  hundred  was  legal  for  balks,  deals,  eggs,  spars, 
stone,  etc. 

Peacock,  in  the  Encyclopaedia  Metropolitana,  I,  381, 
says:  "The  technical  meaning  attached  by  merchants  to 
the  word  'hundred'  associated  with  certain  objects,  was 
six  score — a  usage  which  is  commemorated  in  the  popu- 
lar distich  or  Old  Saw : 

"Five  score  of  men,  money  and  pins, 
Six  score  of  all  other  things." 

Just  so  the  Norwegians  and  Icelanders  have  two  sorts 
of  thousand,  the  lesser  and  the  greater,  the  lesser  =10x 
100,  but  the  greater  =12x100;  and  this  latter  is  called 
tolfraed,  twelfth-red,  a  word  the  exact  analogue  of  our 
hund-red,  tenth-reckoning.* 

All  this  abundantly  proves  that  our  hundred  is  very 
far  from  being  a  simple  numeral  adjective,  like  e.  g., 
seventy;  so  that  while  we  properly  say  seventy-five,  to 
say  a  hundred-five  is  a  hideous  blunder. 

Hundred  is  strictly  not  an  adjective  at  all,  but  a  col- 
lective noun ;  it  is  always  preceded  by  a  definitive,  usually 
an  article  or  a  numeral,  and  if  followed  by  a  numeral, 
this  must  invariably  be  preceded  by  the  word  "and." 

A  following  noun  is,  historically,  a  genitive  partitive, 

*Hickes,  Institutiones  Grammaticae,  p.  43. 


THE  PSYCHOLOGY   OF   READING  A   NUMBER.  97 

in  Old  English  a  genitive  plural,  later  a  plural  preceded 
by  "of."  Thus  1663,  Gerbier,  Counsel,  "About  one 
hundred  of  Leagues."  Hale  (1668)  :  "These  many  hun- 
dred of  years."  Cowper  (1782)  Loss  of  Royal  George: 
"Eight  hundred  of  the  brave."  To-day:  "A  hundred  of 
my  friends,"  "A  hundred  of  bricks,"  "Some  hundreds  of 
men  were  present."  [Murray]. 

Even  if  there  be  an  ellipsis  of  "of"  before  the  noun, 
the  word  hundred  retains  its  substantival  character  so  far 
as  to  be  always  preceded  by  "a"  or  some  adjective.  Com- 
pare "dozen,"  which  has  precisely  parallel  constructions, 
e.  g.,  "a  dozen  of  eggs."  Hooke  (1665)  :  "A  hundred 
and  twenty-five  thousand  times  bigger."  Murray's  Dic- 
tionary (1901)  gives  as  model  modern  English:  "Mod. 
The  hundred  and  one  odd  chances."  Again  it  says:  "c. 
The  cardinal  form  hundred  is  also  used  as  an  ordinal 
when  followed  by  other  numbers,  the  last  of  which  alone 
takes  the  ordinal  form:  e.g.,  'the  hundred-and-first,'  'the 
hundred-and-twentieth,'  'the  six-hundred-and-fortieth 
part  of  a  square  mile.' "  Goold  Brown,  The  Grammar 
of  English  Grammars:  "Four  hundred  and  fiftieth." 

All  this  furnishes  complete  explanation  and  warranty 
of  the  "and"  which  must  always  separate  "hundred" 
from  a  following  numeral.  It  marks  a  complete  change 
of  construction :  "a  hundred  of  leagues  and  three  leagues" ; 
"a  hundred  and  three  leagues."  This  fine  English  usage 
is  unbroken  throughout  the  centuries.  Thus,  Byrhtferth's 
Handboc  (about  1050)  :  "twa  hundred  &  tyn";  Cursor 
MS.  8886  (before  1300)  :  "O  quens  had  he  [Solomon] 
hundrets  seuen."  Myrr.  our  Ladye  (1450-1530)  309: 

"Twyes  syxe  tymes  ten,  that  ys  to  a  hundereth  and 
twenty." 

Oliver  Wendell  Holmes,  "The  Deacon's  Masterpiece" : 


98  FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

"Seventeen  hundred  and  fifty-five. 
Georgius  Secundus  was  then  alive, — 
Snuffy  old  drone  from  the  German  hive." 

The  London  Times  of  February  20,  1885  :  "The  hun- 
dred and  one  forms  of  small  craft  used  by  the  Chinese  to 
gain  an  honest  livelihood." 

The  new  Encyclopaedia  Britannica,  llth  Edition, 
1911,  Vol.  2,  p.  523:  "Thus  we  speak  of  one  thou- 
sand eight  hundred  and  seventy-six,  and  represent  it  by 
MDCCCLXXVI  or  1876."  Again,  p.  526:  "A  set  of 
written  symbols  is  sometimes  read  in  more  than  one 
way.  Thus  1820  might  be  read  as  one  thousand  eight 
hundred  and  twenty  if  it  represented  a  number  of  men, 
but  it  would  be  read  as  eighteen  hundred  and  twenty  if 
it  represented  a  year  of  the  Christian  era." 

Though  all  the  numerals  up  to  a  hundred  belong  in 
common  to  all  the  Indo-European  languages,  the  word 
thousand  is  found  only  in  the  Teutonic  and  Slavonic 
languages,  and  maybe  the  Slavs  borrowed  the  word  in 
prehistoric  times  from  the  Teutons. 

Very  naturally  thousand  is  construed  precisely  like 
hundred :  "Land  on  him  like  a  thousand  of  brick" ;  "The 
Thousand  and  One  Nights." 

And  just  so  it  is  with  that  marvelous  makeshift  mil- 
lion, "big  thousand,"  Old  French  (1359)  augmentative 
(Latin  mille,  a  thousand  +-one  augmentative  suffix). 

Says  Langland  in  Piers  Plowman  (1362)  A,  III, 
255:  ' 

"Coveyte  not  his  goodes 
For  Milions  of  Moneye." 

And  the  divine  Shakespeare  [Henry  V,  Prol.],  antici- 
pating the  telephonic  oh  for  naught: 


THE  PSYCHOLOGY  OF  READING  A  NUMBER.  99 

"Or  may  we  cram 

Within  this  wooden  O  the  very  casques 
That  did  affright  the  air  at  Agincourt? 
O,  pardon!  since  a  crooked  figure  may 
Attest,  in  little  space,  a  million! 
And  let  us,  ciphers  to  this  great  accompt, 
On  your  imaginary  forces  work." 

"Thus,  we  say  six  million  three  hundred  and  twenty 
thousand  four  hundred  and  thirty-six,"*  which  does  not 
at  all  militate  against  our  reading  0033  to  the  telephone 
girl  as  "oh,  oh,  three,  three."  The  word  which  speci- 
fies the  local  value  of  the  digit  is  best  omitted  when  this 
local  value  is  unimportant  or  is  otherwise  determined. 
The  date  1911  read  "nineteen  eleven."  The  approxima- 
tion 77  =  3.14159265+  read  "pi  equals  three,  point,  one, 
four,  one,  five,  nine,  two,  six,  five,  plus."  Here,  as  in  all 
decimals,  the  "point"  fixes  the  local  factor  for  every  sub- 
sequent digit. 

The  country  schoolmaster's  use  of  "and"  solely  to 
indicate  the  decimal  point  is  not  merely  bad  form  and 
stupid;  it  is  criminal.  It  introduces  a  completely  un- 
necessary ambiguity,  doubt,  anxiety  into  the  understand- 
ing even  of  oral  whole  numbers,  since  he  may  end  with 
a  wretched  fractional,  such  as  hundredths,  a  retroactive 
dampener  over  all  that  has  preceded  it. 

When  that  most  spectacular  of  Frenchmen,  who,  like 
so  many  great  Frenchmen,  was  an  Italian,  witness  Maza- 
rin,  Lagrange,  Cassini,  etc,  etc., — when  the  comparatively 
unlettered  Corsican,  Napoleon,  sat  upon  his  white  horse 
at  a  German  jubilee  while  an  official  opened  at  him  an 
address  of  felicitation,  the  great  Captain  began  to  be 
puzzled  at  the  silent  strained  attention  of  those  listeners 
who  were  supposed  to  understand  German  speech.  He 

*  Whitney,  Essentials  of  English  Grammar,  p.  94. 


100        FOUNDATION  AND  TECH  NIC  OF  ARITHMETIC. 

whispered  to  his  aide,  "Why  do  they  not  applaud?" 
"Sire,"  was  the  answer,  "on  attend  le  verbe."  Just  so 
when  the  country  schoolmaster  reads  a  number,  one 
awaits  the  fractional! 

Thus  though  we  may  now  read  Room  203  as  "room 
two-oh-three"  or  as  "room  two,  naught,  three,"  or  as 
"room  two  hundred  and  three,"  reading  it  "room  two 
hundred  three"  remains  an  abominable  gaucherie,  a  nau- 
seating blunder. 


CHAPTER  XVI. 

ARITHMETIC  AS  FORMAL  CALCULUS. 

The  propositions  of  arithmetic,  as  the  body  of  doc- 
trine concerning  numbers  and  certain  operations  by  which 
numbers  may  be  combined,  are  all  deducible  from  a  few 
assumptions. 

In  a  formal  calculus  we  suppose  ourselves  to  know 
nothing  of  the  elements  (represented  by  letters)  or  their 
rules  of  combination  (conventions  by  which  two  elements 
give  a  third)  (represented  by  symbols)  except  our  as- 
sumptions, which  themselves  are  empty  frames  or  forms. 

If  a  specific  meaning  be  read  into  the  letters  and  sym- 
bols, a  true  proposition  may  result,  or  a  false. 

The  logical  deductions  made  from  such  empty  frames 
must  needs  be  formal,  but  this  is  of  advantage  in  keeping 
the  logic  pure  and  unaffected  by  additional  unconscious 
assumptions  which  might  vitiate  it. 

We  propose  to  treat  a  system,  a  Formal  Calculus, 
which  has  arithmetic  as  a  special  interpretation. 

Whatever  has  the  properties  laid  down  in  the  assump- 
tions will  of  necessity  have  also  the  properties  therefrom 
deducible. 

We  shall  set  up  therefore  a  Formal  Calculus  of  which 
Rational  Arithmetic  shall  be  merely  a  true  special  case. 

This  chapter  is  essentially  a  contribution  from  Dr. 
R.  L.  Moore,  of  the  University  of  Pennsylvania. 

Our  elements  are  denoted  by  small  italics,  a,  b,  c. . ... 


102        FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

x,  y,  z,  and  may  for  convenience  be  called  "positive  in- 
tegers," for  which  "integers"  is  only  an  abbreviation. 

Equality  is  denoted  by  =,  inequality  by  ^ ;  the  equality 
of  two  elements  meaning  that  either  is  everywhere  re- 
placeable by  the  other.  Our  two  rules  of  combination 
are  symbolized  by  +  and  x,  and  may  for  convenience  be 
called  addition  and  multiplication. 

I.   1.  If  a  and  b  are  given  elements  (a=b  or  a^b}, 

then  a  +  b  (order  considered)  is  a  uni vocally 
Assumptions.    ,  .      . ,       .  „    ,     „  , 

determined    element    called      the    sum,    a 

plus  b." 

I.  2.  Commutativity :  a  +  b  =  b  +  a. 

I.  3.  Associativity:  a+  (b  +  c)  =  (a  +  b)+c. 

I.  4.  If  a  T*  b,  then  there  is  not  more  than  one  ele- 
ment, z,  such  that  a  +  z  =  b. 

II.  1.  If  a  and  b  are  given  elements  (a=b  or  a^b), 
then  axb  (written  also  simply  ab)    (order  considered) 
is  a  univocally  determined  element  called  "the  product, 
a  by  b." 

II.  2.  Commutativity:  axb  =  bxa. 

II.  3.  Associativity :  ax  (bxc)  =  (ax &)  xc. 

II.  4.  If  axx  =  axy,  then  x=y. 

III.  Distributivity :  ax  (b  +  c)  =  (axb)  +  (axe). 

Theorem  I.  If  m  =  nxx,  then  mnf-m'n  is  a  necessary 
and  sufficient  condition  that  m'-n'x. 

Proof:  Firstly  if  m  =  nx*  and  m'=n'x,  then  m(n'x}  - 
mm'. 

Hence,  by  II  3,  (ww'):r=  (nx)mf; 
by  II  2,  (mn'}x=m'(nx)  ; 
by  II  3,  (mn'')x=  (m'ri)x\ 

*  The  sign  X  between  two  letters  will  hereafter  often  be  omitted 
and  understood. 


ARITHMETIC  AS  FORMAL  CALCULUS.  103 

by  II  2,  x(mri}-x(m'ri)\ 
by  II  4,  mn'=m'n. 

Conversely,  if  m  =  nx  and  mn'  =  m'n,  then  (w'n)w  = 
(ww')  (TUT). 

Hence,  by  II  2,  II  3,  (mw)w'=  (WM)  (w'*) ; 
by  II  4,  w'  =  w'.*r 

Definition  1 :  Hm  =  nx,  then  and  only  then  x=m/n. 

If  for  a  certain  pair  of  integers,  m  and  w,  there  is  no 
integer  x  such  that  m-nx,  and  thus  no  integer  equal  to 
m/n,  then  if  one  wishes  that  in  this  case  also  there  should 
be  something  which  is  equal  to  m/n,  that  m/n  should 
enter  our  Formal  Calculus  as  a  new  kind  of  element,  he 
may  choose  something  other  than  an  integer  which  it 
would  be  convenient  to  call  m/n.  He  is  at  liberty  in  this 
case  to  call  anything  (except  an  integer)  m/n.  Such  a 
definition  could  never  possibly  contradict  our  assump- 
tions or  previous  definitions  since  according  to  hypoth- 
esis there  is  no  x  such  that,  in  sense  of  previous  Defini- 
tion 1,  m/n=x.  Now  what  shall  we  in  this  case  call 
"w/w"?  It  is  desired  to  establish  a  Formal  Calculus 
which  shall  contain  ordinary  arithmetic.  It  is  desired 
then  that  m/n,  in  this  case  also,  and  operations  in  which 
it  is  to  figure,  should  be  such  that  certain  laws  may  be 
obeyed.  One  thing  which  is  desirable  then  is  that  (as 
in  the  case  when  m/n  is  an  integer)  m/n  and  w'/w'  here 
also  shall  mean  the  same  thing  only  in  case  mn'  =  m'n. 
What  sort  of  definition  for  m/n  would  satisfy  this  con- 
dition ? 

Evidently  the  following  does: 

Definition  2:  If  there  is  no  x  such  that  m-nx,  then 
m/n  means  the  set  of  all  sensed  pairs  (p,q)  such  that 
mq  =  pn. 


104       FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

For:  Theorem  2:  If  mq  =  pn,  then  mn'  =  m'n  is  a  ne- 
cessary and  sufficient  condition  that  m'q  =  pn'. 

Proof:  If  mn'  =  m'n  and  mq  =  pn  then  (mq}(mfn}- 
(pn}(mn'}.  Hence,  by  II  2  and  II  3,  (mn}(m'q}  = 
(mn}  (pn'}. 

From  Definition  2  and  Theorem  2  it  is  seen  that  if 
there  is  no  x  such  that  m  =  nx,  then  mn'  =  m'n  is  a  neces- 
sary and  sufficient  condition  that  m/n  =  m' '/n' ';  this  is 
easily  seen  except  perhaps  for  an  obstacle  which  one  may 
indicate  thus :  Suppose  it  should  happen  that,  even  though 
mn'  =  m'n  and  there  is  no  x  such  that  m  =  nx,  still  there 
is  an  x  such  that  m'  =  n'x  and  thus  m' /n'  may  not  be  the 
set  which  m/n  is  according  to  Definition  2.  But  if  mn'- 
m'n,  and  there  is  no  x  such  that  m  =  nx  then,  by  Theorem 
1 ,  there  can  be  no  x  such  that  mf  =  n'x. 

From  this  result  and  Theorem  1  we  have  finally: 

Theorem  3 :  In  any  case  mn'  —  m'n  is  a  necessary  and 
sufficient  condition  that  m/n  =  m'/n'. 

Theorem  4:  If  m-nx  and  m'^n'x*  then  1°  m/n  + 
m' /n'  =  (  mn'  +  nm'}  /nnf  and  2°  m/n  x  mf  /n'  =  mm'/nn'. 
Proof:  1°.    mn'  +  nm'=  (nx}n' +  n(n'xr}. 
Hence,  by  II  2,  II  3,  and  III. 

m  n'  +  nmf  =  nnf  (  x  +  x' ) . 
Hence,  by  Definition  1 

x+x*=  (mn'+nm'}/nnf. 
2°.  mm'/nn'—  (nx}  (n'x'^/nn'. 
Hence,  by  II  2,  II  3, 

mm'/nn'=  (nn')  (xxf}/nn'. 
Hence,  by  Theorem  3, 

(mm'}  (nn')  =  [  (nn'}  (xx'}  ]  (nn'}. 
Hence,  by  II  2  and  II  4, 

mm'—  (nn'}  (xx'}. 


ARITHMETIC  AS  FORMAL  CALCULUS.  105 

Hence,  by  Definition  1, 
mm'/nn'  =  xxf. 

Definition  3  :  If  for  any  particular  integers  m,  n,  m', 
n',  either  there  is  no  x  such  that  m  =  nx  or  there  is  no  x' 
such  that  m'-n'x',  then  1°.  m/n  +  m'/n'  means  (mn'  + 
mn')/nn'  and  2°.  m/nxm'/n'  means  mm'/nn'. 

In  order  that  there  should  be  no  contradiction  here, 
in  order  that  this  may  not  be  defining  one  thing  as  being 
the  same  as  two  different  things,  it  is  necessary  that  this 
following  theorem  should  be  true  : 

Theorem  4:  If  m/n  =  a/b,  and  m'/n'-a'/b',  then  1°. 
(mn'  +  nm'}/nn'  =  (ab'  +  ba')/bb';  and  2°.  wm'/nn'- 
aa'/W. 

Proof:  l°:By  hypothesis  and  Theorem  3,  mb  =  an 
and  m'b'  =  a'n'.  Hence, 


Hence,   (mb)  (&V)  +  (m'b'}  (bn)  =  (an)  (6V)  + 

(aV)(fe«). 
Hence,  by  II  2,  II  3,  III,  (mn'+nm')bb'  =  nn'(ab'  + 

ba'Y 
Hence,  by  Theorem  3,   (wn'  +  ww')/«w'=  (ab'  + 

ba')/bb'. 
2<>.   From  mb  =  an  and  m'b'=a'n'  it  follows  that 

(w&)(m/&/)  =  (aw)(aV). 

Hence,  by  II  2  and  II  3,  (mm'}  (bb'}  =  (aa')  (nnr). 
Hence,  by  Theorem  3,  mm'/nn'=aa'/bb'. 

Theorem  5:  In  any  case,  m/n  +  m'/n'  =  (mn'  +  nm')/ 
nn',  and  (m/nxm'/n')=mm'/nn'. 

Proof  :  See  Theorem  4  and  Definition  3. 


106        FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

Definition  4:  If  m  and  n  are  any  two  integers  (the 
same  or  different)  then  m/n  is  called  a  positive  fraction, 
for  which  fraction  is  only  an  abbreviation.  Conversely, 
every  fraction  is  m/n,  where  m  and  n  are  integers  (the 
same  or  different). 

Theorem  6:  Every  integer  is  a  fraction. 

Proof'.  If  m  is  an  integer,  then,  by  Definition  1, 
mm/m=m.  Hence,  by  II  2  and  Definition  4,  m  is  a 
fraction. 

Capital  letters  are  used  here  to  designate  fractions 
only. 

Theorem  7 :  If  A,  B,  C  are  fractions,  then  the  follow- 
ing statements  are  true : 

F  1.    A  +  B  and  AxB  are  fractions. 

F  2.    A+B  =  B  +  Aand  AxB  =  BxA. 

F  3.    (A  +  B)+C  =  A+(B  +  C),  and  (AB)C  = 

A(BC). 
F  4.    There  is  not  more  than  one  fraction,  D, 

such  that  A  +  D  =  B,  and  there  is  not 

more  than  one  fraction  E  such  that 

AxE-B. 

F  5.  There  is  a  fraction  F  such  that  AxF  =  B. 
F  6.    There  is  a  fraction  G  such  that,  if  H  is 

any  fraction  whatsoever,  then,  GH  =  H. 
F  7.   A(B  +  C)=AB  +  AC 

Proof  of  F  1 :  See  Theorem  5,  Definition  4,  I  1  and 
II  1. 

Proof  of  F  2 : 

a.  By  Theorem  5,  m/n  +  m'/n'  =  (mn'  +  nm' )/««'. 


ARITHMETIC  AS  FORMAL  CALCULUS.  107 

Hence,  by  I  2  and  II  2,  m/n  +  m'/n'=  (m'n  +  n'm)/ 

nn'. 
Hence,  by  Theorem  5,  m/n  +  m'/n'=m'  /n'  +  m/n. 

b.    By  Theorem  5,  (w/wxw'/V)  =mm'/nnf, 
Hence,  by   II   2   and   Theorem   5,    (ra/nxw'/n')  = 


Proof  of  F  3  : 

a.    By  Theorem  5,  m/n+  (m'/n'  +  m"/n") 
=  m/n  +  (  m'n"  +  n'm"  )  /n'n" 
=  O(Vtt")  +  ttO'n"  +  w'm")]/«(w'«"),  which 

by  HI, 
=  [w(w'w")  +  »(ro'n")  +  «(»'w")]/»(«'n"), 

which  by  II  3, 
=  [(mw')w"  +  (nw')»"  +  (nn')m"]/(nn')n", 

which  by  II  2  and  III, 


=  (  mw'  +  nm')  /nn'  +  m"  /  n"  =  (  m/n  +  m'/n'  )  + 
m"/n". 

b.  By  Theorem  5  and  II,  3, 

m/nx  (W'/M'XW"/W")  =m/nx  (m'm"  '/n'n") 
=  m(m'm")/n(b'n")  =  (mm'}m"  /  (nn')n" 
=  (W/MXW'/W')  xmr//w//- 

Proo/  of  F  4  : 

a.  If  a/b  +  x/y  =  c/d  and  a/b+x'/y'  =  c/d,  then,  by 
Theorem  5  ,  (  <ry  +  for  )  /fry  =  (  a/  +  fo/  )  /by*. 

Hence,  by  Theorem  3,  II  2  and  II  3, 
(W)  (^y)  =  (ay)  (&/)  +  (&&)  (^). 

Hence,  by  I  4,  (&&)  (^/)  =  (&&)  (^3;). 

Hence,  by  II  4,  xtf^-tfy. 

Hence,  by  Theorem  3,  x/y= 


108        FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

b.  If  (a/bxx/y)=c/d  and  (a/bxx'/y'}=c/d,  then 
by  Theorem  5,  ax  /by  -  ax*  /by'. 

Hence,  by  Theorem  3,  II  3  and  II  2,  (ab)  (xy')  = 
' 


Hence,  by  II  4,  xy'-x'y. 

Hence,  by  Theorem  3,  x/y  =  x'/y'. 

Proof  of  F  5  : 

A  =  m/n  and  B  =  m'/w',  then  Ax  (m'n/n'm)  =m/n 
xm'n/n'm,  which,  by  II  2  and  II  3,  =  (mn)m'/(mn*)n'. 
But,  by  II  2  and  II  3,  [(WW)W']W'  =  W'[(WM)W']. 
Hence,  by  Theorem  3,  (mri)m'/  '  (mn)n'  =  m'/n'. 
Thus  Ax  (m'n/n'm)  =B. 

Proof  oi  F  6: 

If  H  =  m/n,  and  ^  is  any  integer,  then  by  Theorem  5, 


But,  by  II  2  and  II  3,  (km~)n  =  m(kn). 

Hence,  km/kn  =  m/n. 

Hence,  for  any  H  whatever,  (&/£)H  =  H. 

Proof  of  F  7: 

By  Theorem  5  ,  m/n  x  (  mr  /  n'  +  m"/n"  )  =  m/n  x  [  (  m'n" 
+  n'm")/n'n"'\=  mx(m'n"  +  n'm")/[nx(n'n")],  which 
by  Theorem  3,  II  2  and  II  3,  =  [(mm')  (nn"}  +  (nn') 
(mm")]/(nn')  (nn"),  which  by  Theorem  5,  =(m/nx 
wi'/ri  )  +  (  m/n  x  m"  /n"  )  . 

Definition  : 

m/n>  m'/nf  means  there  exist  x,  y  such  that  m/n- 
m'/n'+x/y.  m/,n  <  m'/ri  means  mf  /n'  >  m/n. 

Assumption  IV:  A  necessary  and  sufficient  condition 
that  integer  a  should  be  different  from  integer  b  is  the 


ARITHMETIC  AS  FORMAL  CALCULUS.  109 

existence  of  an  integer  x  such  that  either  a  +  x=b  or 


If  this  assumption  IV  is  added  to  the  others,  then 
the  following  additional  statements  may  be  added  in  The- 
orem 7: 

F  8.  Either  A  <  B,  A  =  B,  or  A  >  B.  But  no  two 
of  these  three  statements  are  simultaneously  true. 

F  9.*  If  A  >  B,  and  B  >  C,  then  A  >  C 

F  10.  If  A>B,  then  A  +  OB  +  C,  and  AC>  BC 

*  F  9  and  F  10  may  be  proved  without  use  of  IV. 


CHAPTER  XVII. 

ON  THE  PRESENTATION  OF  ARITHMETIC 
FIRST  GRADE. 

All  schools  heretofore  have  commenced  the  study  of 
number  by  asking  and  considering  the  answers  to  the 
Previous  questions,  "how many?"  "how much?"  "how 
blunders.  far?"  «how  iong?"  They  have  thus  begun 
with  the  cardinal,  and  with  it  alone  have  continued.  Thus 
all  teaching  of  the  beginnings  of  arithmetic  has  uncon- 
sciously overlooked  and  missed  the  more  fundamental 
and  prerequisite  question,  "which  one  ?",  and  so  remained 
unconscious  of,  and  blind  to  the  infinitely  precious  and 
in  fact  indispensable  succor  and  aid  of  order,  of  the 
ordinal. 

Had  study  of  the  child  been  fructified  by  foreknowl- 
edge of  the  modern  higher  mathematics,  it  could  not  have 
Begin  with  overlooked  in  the  spontaneous  creative  ac- 
ordinals.  tivities  of  the  child,  the  prominence  and 
absolutely  basal  character  of  the  ordinal,  non-cardinal 
ideas,  the  serial,  arranging  and  identifying  ideas,  histor- 
ically and  developmentally  preceding  and  prerequisite  for 
the  very  apparatus  subsequently  used  for  the  ascertain- 
ment of  the  "how  many." 

In  the  counting  of  a  primitive  group,  any  element  is 
considered  equivalent  to  any  other.  But  in  the  use  even 
of  the  primitive  counting  apparatus,  the  fingers,  appeared 
another  and  extraordinarily  important  character,  order. 


ON  THE  PRESENTATION   OF  ARITHMETIC.  Ill 

The  savage,  in  counting,  systematically  begins  his 
count  with  the  little  finger  of  the  left  hand,  thence  pro- 
ceeding toward  the  thumb,  which  is  fifth  in  the  count. 
When  number-words  come  to  serve  as  extended  counting 
apparatus,  order  is  not  only  a  salient  but  an  absolutely 
essential  and  indispensable  characteristic  of  the  apparatus. 
The  number  series,  1,  2,  3,  and  so  on,  is  a  system  such 
that  for  every  element  of  it  there  is  always  one  and  only 
one  next  following. 

Numbers  are  ordinal  as  individuals  in  a  well-ordered 
set  or  series,  and  used  ordinally  when  taken  to  give  to 
any  one  object  its  position  in  an  arrangement  and  thus 
individually  to  identify  and  place  it. 

The  ordinal  process  has  also  as  outcome  knowledge 
of  the  cardinal.  When  we  have  in  order  ticketed  the 

_  ,.  .  ninth,  we  have  ticketed  nine.  Thus  the  last 
Cardinal 

from  ordinal  used  tells  the  result  of  the  count. 

ma '  But  this  very  ordering  process  precedes  all 

cardinal  ideas,  as  is  shown  by  that  use  of  count  which 
occurs  in  the  spontaneous  games  of  little  children,  in 
their  counting  out  or  counting  to  fix  who  shall  be  it. 

This  counting  is  characterized  by  order  pure  and 
simple.  There  is  successive  designation  with  no  attempt 
Ordinal  a*  simultaneous  apprehension,  simply  the  as- 

countmg.  signment  of  order  to  a  collection  and  the 
ascertainment  of  place  in  the  series  made  by  this  putting 
in  order.  Our  instrument  for  this  is  the  number  series, 
and  it  is  upon  the  order  in  the  system  that  we  ourselves 
rely  to  get  a  working  hold  of  the  individual  number, 
especially  when  beyond  the  point  where  we  can  have  any 
complete  appreciation  of  the  simultaneous  multiplicity  of 
the  units  involved  in  the  corresponding  cardinal. 


112        FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

It  is  fortunate  then,  and  natural,  that  the  modern 
child,  despite  the  blindness  of  its  teachers  hitherto,  gets 
the  words  of  the  ordinal  series  before  it  gets  the  cardinal 
concepts  we  attach  to  them. 

The  ordinal  coherence  of  the  number  series  and  its 
independence  of  cardinal  concepts  is  shown  by  the  child. 
Each  name  depicts  a  natural  individual,  not  the  so-far 
group  of  natural  individuals,  not  a  new  kind  of  unity 
composed  of  units. 

Our  apparatus  for  the  ascertainment  of  cardinal  num- 
ber involves,  is  based  upon  and  uses  order,  ordinal  num- 
Cardinal  her.  The  child  should  be  counting  up  to 
counting.  a  hundred  before  it  can  recognize  a  group 
of  seven  objects.  When  the  symbols  of  the  number  series, 
the  natural  scale,  are  mated  in  sequence  with  the  elements 
of  an  aggregate,  the  last  symbol  used  is  also  taken  as 
designation  of  the  particular  whole  set  so  far  used,  and 
this  identification  of  the  unknown  set  with  a  known  set 
it  is  which  gives  the  cardinal  property  or  quality  of  the 
hitherto  unknown  set.  In  this  sense  we  say  the  last 
symbol  used  gives  the  outcome  of  the  count,  tells  the 
cardinal  number  of  the  counted  aggregate. 

First  of  all  then  let  the  teacher  put  out  of  her  mind 
the  blunder,  pedagogic  as  well  as  scientific,  that  number 
N  .  was  in  any  way  dependent  upon  measure- 

precedes  ment  for  origin.  Number  was  created  and 
used  for  individual  ordering  and  identifica- 
tion and  for  group  identification  centuries  before  any 
measurement.  There  are  tribes  now  using  number  that 
never  have  used  measurement.  All  natural  children  use 
number  long  before  measurement  can  even  be  explained 
to  them.  Measurement  is  a  recondite  device.  Number 
is  enormously  more  simple  and  primitive.  Its  uses  in 


ON  THE  PRESENTATION   OF  ARITHMETIC.  113 

identification  both  of  individuals  and  groups  are  vastly 
important  and  quite  independent  of  measurement.  They 
long  precede  any  thought  of  measurement. 

The  number  concepts  are  wholly  apart  from  measure- 
ment, from  length,  from  size,  from  the  late-coming  con- 
ventional standards  for  measurement,  from  the  yard, 
the  mile,  the  grain,  the  liter  or  any  other  standard  for 
measurement.  Valuation  is  a  false  associate  for  primi- 
Cardinal  tive  number.  Number  implies  no  exact  size 
number.  image.  Cardinal  number  is  a  quality  of  a 

group.  Two  eyes  and  an  ear-ache  is  a  less  dangerous 
trio  than  three  yards,  lest  the  teacher  make  the  mistake 
of  supposing  number  in  any  way  dependent  upon  meas- 
urement. 

It  is  the  acme  of  stupidity  to  attempt  to  found  the 
number  concepts  upon  "how  much";  for  example,  "my 
desk  is  greater  in  length  than  in  width." 

Begin  by  letting  the  child  sing  the  number  names  as 
far  as  it  enjoys  the  singing.  Follow  this  up  by  exercises 
How  to  m  designating  or  tagging  objects  with  these 

begin.  number-names  as  identifying  tags.  Paper 

horses  may  be  used,  named  one,  two,  three,  etc.  Paper 
automobiles  may  be  named,  as  the  real  ones  are  tagged, 
one,  two,  three,  etc.  Objects  so  tagged  may  be  jumbled 
up  and  then  arranged  in  the  order  of  their  names.  Then 
differing  objects,  say  the  various  differing  animals  in 
animal  crackers,  may  be  named,  each  with  a  number. 
Then  the  qualities  of  No.  2  may  be  contrasted  with 
those  of  No.  4. 

The  children  may  each  be  given  a  number  as  name. 
The  teacher  and  the  children  may  invent  games  using 
the  ordinal  properties,  carefully  avoiding  as  yet  any 
"how  many." 


1 14        FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

(1)  Thus  an  instructive  ordinal  game   is   using  a 
set  of  ordinals  to  count  out  the  class.     Choose  a  set 
Ordinal  °f  ordinals,  say  the  first  nine.     Distribute 
games.  them  in  order  and  let  the  child  to  whom 
the  nine  comes  be  out.     Then  begin  again  with  the  re- 
maining children  and  again  distribute  the  ordinals  in 
order,  dropping  as  out  the  child  upon  whom  the  nine  now 
falls.     When  there  are  only  eight  children  remaining, 
the  count  will  more  than  go  around,  and  the  child  tagged 
with  one  will  also  be  tagged  with  nine  and  so  be  out,  etc. 

(2)  Give  each  child  the  same  set  of  disarranged  num- 
bers.   See  who  can  arrange  quickest. 

(3)     One,  two; 
Buckle  my  shoe. 

Three,  four; 
Open  the  door. 

Five,  six; 
Pick  up  sticks. 

Seven,  eight; 
Lay  them  straight. 

Nine,  ten; 
A  big,  fat  hen. 

Eleven,  twelve; 
Dig  and  delve. 

(4)     One,  two,  three,  four,  five; 
I  caught  a  bird  alive. 
Six,  seven,  eight,  nine,  ten; 
I  let  it  go  again. 

The  call.         (5)   One,  two; 

Glad  to  see  you. 
Three,  four; 


ON  THE  PRESENTATION   OF  ARITHMETIC.  115 

Open  the  door. 

Five,  six; 

My  dog  does  tricks. 

Seven,  eight; 

Walk  to  the  gate. 

Nine,  ten ; 

Please  come  again. 

(6)  Mix  up  nine  blocks  numbered  from  one  to  nine. 
Let  the  child  draw  them  out  of  the  heap  and  put 

down  each  in  its  relative  place  when  drawn  until  all 
are  arranged  in  their  proper  order. 

(7)  Hang  about  the  neck  of  each  of  nine  children  a 
numbered  tag.   Let  the  children  arrange  themselves  in 
order  in  line.  Bend  the  line  into  a  closed  curve.  Call  out 
one  number.  The  child  so  numbered  goes  within  the  en- 
closure.    The  others  march  about  him.     At  a  signal  he 
calls  a  number.     The  child  so  designated  takes  his  stand 
within  the  encircling  line,  and  the  caller  finds  his  proper 
place  in  the  line. 

(8)  Give  a  number  to  each  animal  in  a  Noah's  ark. 
This  so  far  is  only  a  nominal  number,  a  name  for  a 
natural  individual.     Then  introduce  the  ordinal  by  let- 
ting the  child  arrange  the  numbered  animals  in  accord- 
ance with  their  number-names.    Animal  crackers  may  be 
substituted  for  a  Noah's  ark. 

(9)  Have  colored   strips  of  paper  numbered  con- 
secutively in  correspondence  with  the  colors  in  the  pri- 
mary rain-bow.    Let  the  children  arrange  them  in  order 
to  make  a  rain-bow. 

(10)  Shuffle  a  pack  of  numbered  cards.     Give  the 
pack  to  the  child  to  arrange  in  the  order  of  the  numbers. 


116        FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

(11)  Let  the  aisles  in  the  school-room  be  numbered 
streets,  and  the  broad  cross  passage-ways  numbered  ave- 
nues, and  each  desk  a  numbered  house.  Let  the  children 
write  and  address  notes  giving  the  house  address,  and 
let  a  messenger-child  carry  and  deliver  the  letters. 

Addition : 

Ordinal  In  the   ordered   row   of   children   ask: 

operations.      Which  is  the  third  after  the  second?    An- 
swer: the  fifth. 

Subtraction : 

Which  is  the  third  before  the  fifth?  Answer:  the 
second. 

Multiplication : 

Which  is  the  third  second?    Answer:  the  sixth. 

Which  is  the  second  third?    Answer:  the  sixth. 

When  the  child  is  thoroughly  familiar  with  the  or- 
dered names  as  applied  to  natural  individuals,  we  are 
The  simplest  ready  for  their  first  application  to  artificial 
cardinal.  individuals  of  the  group  kind,  and  first  the 
application  of  the  ordinal  two  to  a  pair.  Make  couples, 
partners,  pairs,  mates,  and  call  each  pair  two. 

The  cardinal  two,  the  simplest  cardinal,  is  that  prop- 
erty of  a  set  whereby  it  can  be  mated,  one  to  one,  with 
a  child's  thumbs,  or  it  is  the  class  of  such  sets. 

The  idea  of  a  cardinal,  belonging  as  it  does  to  a  set 
of  things  as  a  whole,  is  a  comparatively  late  concept.  It 
must  follow  the  concept  of  a  whole  composed  of  parts, 
constituents  permanently  distinguishable.  Later  comes 
the  attribution  of  the  geometirc  quality  of  relative  size, 
big  and  little,  to  numbers. 

For  the  next  step  make  trios.    The  cardinal  three  is 


ON  THE  PRESENTATION  OF  ARITHMETIC.  117 

the  class  of  all  triplets,  or  that  quality  of  a  set  whereby 
Triplets  and  ^  can  be  mated,  one  to  one,  with  a  child's 
quartets.  eves  an(j  nOse,  also  with  the  ordinal  set  one, 
two,  three;  the  last  of  which  is  used  as  a  tag  or  name 
for  the  group,  the  trio. 

Quartets  are  groups  mateable,  individual  to  individual, 
with  the  fingers  of  the  left  hand,  or  the  words  one,  two, 
three,  four;  the  last  of  which  is  to  be  used  as  a  name  for 
every  such  set ;  and  so  on.  There  may  follow  in  rich  variety 
the  construction,  the  identification,  the  tagging,  of  small 
The  "how  groups.  This  is  at  last  the  "how  many" 
many"  idea.  jdea  Let  jt  first  be  the  natural  and  useful 
question  of  simple  identification  of  groups,  recognition 
of  like  or  unlike  cardinal. 

Herein  lies  abundant    opportunity   for  constructive 
work.     Give  the  child  the  first  five  ordinals.     Let  him 
then  construct  groups  whose  name  shall  be  five,  conse- 
quently whose  "how  many"  shall  be  five,  the  cardinal. 
Explain  how  simple  groups  were  used  as  symbols 

for  the  numeric  quality  of  all  like  groups. 

Thus,  II,  III,  IIII,  are  symbols  for  their 
own  cardinal  quality  two,  three,  four.  Then  may  come 
the  Hindu  symbols  2,  3,  4,  primarily  as  ordinals,  then 
Cardinal  secondarily  as  cardinals.  Now  is  the  time 
counting.  for  cardinal  counting,  counting  as  group- 
identification,  using  first  the  ten  different  groups  of  fin- 
gers as  known  groups  with  one  of  which  the  unknown 
group  is  to  be  identified  by  setting  up  a  one-to-one 
correspondence  between  the  individuals  of  the  unknown 
R  niti  £rouP  and  the  individuals  of  a  finger  group, 
of  the  car-  Then  we  go  to  cardinal  counting  using  the 

first  dozen  groups  of  ordinal  words  as 
known  groups.  All  in  good  time,  a  test  that  the  idea  of 


1 18        FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

the  cardinal  has  taken  root  and  germinated,  is  practice 
in  the  instantaneous  recognition  of  the  cardinal  of  a  small 
group  suddenly  exhibited,  then  veiled.  The  question 
"how  many"  is  to  be  answered  without  conscious  count- 
ing. Then  larger  groups  may  be  used  recognizable  by 
use  of  symmetry  in  the  arrangement  or  grouping,  as  on 
playing  cards. 

Then  we  may  begin  to  train  for  the  instantaneous 
recognition  from  two  components  of  the  cardinal  of  their 
Cardinal  compound,  for  example,  the  thinking  of 
addition.  seven  upon  seeing  three  and  four.  Here 
we  should  stop  to  train  until  every  pair  from  one  plus 
one  up  to  nine  plus  nine  arouses  the  image  of  its  sum 
instantly  and  automatically.  Coins,  cents,  nickels,  dimes, 
dollars,  are  admirably  adapted  at  this  stage  as  anchors 
for  the  ideas  created,  while  at  the  same  time  bringing 
home  to  the  child  the  precious  aid  of  number  (anterior 
to  any  measurement)  in  the  child's  social  relations,  in 
the  interest  growing  out  of  and  attaching  to  the  very 
life  of  the  child  itself.  Games  of  buying,  and  perhaps 
actual  buying,  with  the  consequent  paying  and  change- 
making,  are  here  in  place. 

Constructive  processes  familiarize  and  endear  to  the 
child  the  ideal  numeric  creations. 

Summary  (First  Grade). 

A.  Ordinal  counting.    Utilize  the  spontaneously  child- 
Ordinal  arith-  create(^  ordinal  systems.     Also  rhymes  and 
metic,  then      jingles, 
cardinal.  R    The  number  symbolS)    1,   2,   3,   4,   5,   6 

etc.  as  ordinals. 

C.  Ordinal  applications,  identification,  arrangement, 
factitious  order. 


ON  THE  PRESENTATION  OF  ARITHMETIC.  119 

D.  Ordinal  tagging. 

E.  Ordinal  games. 

F.  Group-making,  group  distinction,  group  familiari- 
zation. 

G.  Group  identification,  cardinal  counting. 
H.  Cardinal  applications. 

I.    Cardinal  games. 

J.  The  number  symbols  as  cardinals. 

K.  Positional  notation  for  number. 

L.  Addition  tables. 

M.  Coins  and  their  applications  and  games. 

N.  Exercises  in  making  conscious  the  number-needs 
of  the  child's  own  life,  individual  and  social. 

O.  Problems  oral  and  motor ;  ordinal ;  to  be  solved  by 
ordinal  identification  and  arrangement.  Cardinal;  to  be 
solved  by  cardinal  identification,  by  addition,  by  corre- 
lation. 

SECOND  GRADE. 

The  number  work  of  the  second  grade,  as  in  all 
grades,  is  to  be  related  as  closely  as  may  be  to  the  actually 
existing  interests  and  immediate  needs  of  the  child. 

Do  not  bend  for  a  moment  to  the  false  and  exploded 
idea  that  number  was  originated  or  created  by  measure- 
Measure-  ment,  a  palpable  absurdity,  since  we  must 
ment.  already  be  able  to  count  before  we  can  meas- 

ure, and  since  the  preexistent  counting  is  absolutely  exact 
while  no  measuring  ever  can  be  exact.  But  now  that  the 
child  has  the  prerequisite  number-equipment,  we  may 
envisage  measurement. 

The  "muchness"  of  a  quantity  is  not  determined  by 
the  "how  many"  parts  in  it,  unless  these  be  all  of  a 
fixed,  a  preestablished  size.  Hence  in  addition  to,  and 


120        FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

outside  of  the  number-ideas,  the  child  must  now  be  con- 
fronted with  the  new  and  difficult  idea  of  definite  con- 
ventional standards,  the  so-called  units-for-measure  or 
units  of  measure,  the  inch,  the  quart,  the  pound,  the 
second,  the  degree.  To  measure  is  to  break  the  thing 
up  into  pieces  each  equal  to  one  of  these  standards,  or  a 
like  standard,  and  to  count  the  pieces.  The  child  must 
combine  his  old  knowledge  of  the  number  obtained  with 
his  new  knowledge  of  the  standard  now  used. 

Measurement  then  can  only  come  after  much  prac- 
tice in  counting.  Finally  begin  measuring  by  measur- 
ing a  length.  Show  that  nothing  would  be  gained 
here  by  actually  breaking  off  the  pieces,  as  we  do  in 
measuring  milk.  We  need  only  see  where  they  could 
be  broken  off.  Now  we  are  ready  for  the  consideration 
of  the  actual  problems  presented  to  the  child  by  its  own 
occupations.  It  may  be  called  upon  to  use  so  much 
rope  or  board  or  food.  The  outcome  of  the  measure- 
ment is  a  graphic  description  in  known  terms,  a  num- 
ber and  a  unit;  and  now  inversely  a  metric  description 
should  evoke  a  graphic  image,  a  picture. 

Since  mensuration  is  combined  with  arithmetic,  there 
may  be  training  to  familiarize  the  various  units  and  their 
subunits,  yard,  foot,  inch,  gallon,  quart,  pint,  hour,  min- 
ute, second,  pound,  ounce,  gram,  etc. 

Now  should  be  given  a  thorough-going  presentation 
of  our  positional  notation  for  number,  and  as  the  neces- 
sary extension  of  it,  the  decimal.    Decimals 
The  decimal.  ,  ..    ,          ...        .    , ,      , 

are  made  up  of  the  subunits  inevitably  des- 
ignated by  the  extension  of  our  positional  notation  to  the 
right  of  the  units'  column. 

As  the  self-interpreting  extension  of  this  positional 
notation  for  number  to  the  right  of  the  units'  column, 


ON  THE  PRESENTATION  OF  ARITHMETIC.  121 

we  have  decimals.  We  need  no  new  elements,  nothing 
but  the  already  mastered  digits,  base,  column.  The  deci- 
mal is  not  a  fraction;  it  has  no  denominator.  Decimals 
are  significant  figures  to  the  right  of  the  units'  column; 
to  indicate  units'  column,  we  henceforth  use  the  decimal 
point.  One  thousand  (1000)  means  ten  of  such  units 
as  stand  in  the  adjacent  column  to  the  right ;  and  one  of 
these,  one  hundred  (100),  means  ten  of  such  as  stand  in 
the  next  column ;  and  one  of  these,  ten  ( 10) ,  means  ten  of 
our  primal  units,  such  as  stand  in  our  units'  column ;  and 
one  of  these,  One  (1 ),  means  ten  of  such  as  stand  in  the 
next  column  to  the  right,  that  is  in  the  first  column  to  the 
right  of  our  units'  column ;  and  one  of  these,  one-tenth,  .  1 , 
has  the  same  relation  to  one  in  the  next  column.  We  have 
an  excellent  available  illustration  in  our  coins.  Taking  the 
dollar  as  the  primal  unit,  one-tenth,  .1,  is  one  dime  or 
ten  cents;  .01  is  one  cent,  or  ten  mills.  These  columns 
are  to  be  named  so  that  units'  column  be  axis  of  sym- 
metry; twenty  (20)  gives  tens;  so  0.2  gives  tenths; 
three  hundred  (300)  gives  hundreds;  so  0.03  gives  hun- 
dredths ;  then  4000  gives  thousands ;  so  0 . 004  gives  thou- 
sandths. 

As  no  new  elements  come  with  decimals,  nothing 
but  our  old  digits,  base,  column,  so  no  new  principle  is  in- 
volved in  their  addition,  subtraction,  multiplication  and 
division.  The  child  who  has  the  equipment  for  inter- 
preting 23  has  that  for  interpreting  3.14159265.  Our 
_  .  explanation  of  positional  notation  contains 

the  explanation  of  "carrying"  in  addition. 
Whenever  the  digit  X  is  reached  in  any  column,  it  is 
carried,  appearing  as  one  in  the  next  column  to  the  left. 

So  we  have  this  word  already  available  when 
Subtraction.  .  * . 

we  reach  subtraction,  which  is  always  to  be 


122        FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

worked  by  addition.  Look  upon  difference  as  the  num- 
ber which  if  added  to  the  subtrahend  gives  the  minuend. 
Thus  to  subtract,  9004 

5126 

3878 

Think  six  and  eight  make  fourteen ;  carry  1 ;  three  and 
seven  make  ten ;  carry  1 ;  two  and  eight  make  ten ;  carry 
1 ;  six  and  three  make  nine.  We  carry  one  to  balance 
a  one  put  in  to  facilitate  our  procedure.  Thus  in  sub- 
tracting, 

8%      say  two-fifths  and  four- fifths  make  six-fifths; 
6%       carry  1 ;  seven  and  one  make  eight. 
~I%~ 

A  fraction  is  an  ordered  number-pair  where  the  sec- 
ond number,   the   denominator,   tells  what 
Fractions.  ...  11.10 

sort  of  units  are  represented  by  the  nrst 

number,  the  numerator.  Thus  2/3  means  two  of  such 
units  (subunits)  that  three  of  them  make  the  primal  unit. 
When  we  come  to  multiplication,  the  idea  of  column 
Multiplied-  is  to  dominate.  The  fundamental  admoni- 
tion. tjon  js .  Always  keep  your  columns.  Always 
begin  to  multiply  with  left-most  figure  of  the  multiplier. 
Thus  we  get  the  most  important  partial  product  first. 
Rule :  The  figure  put  down  stands  as  many  places  to  the 
right  or  left  of  the  digit  multiplied  as  the  multiplier  is 
from  units'  column. 

21 .354  Another  form  of  the  rule  is :  Mul- 

200 . 003  tiplying  shifts  as  many  places  right 

4270.8  or    left    as    the    multiplier    is    from 
64062       units'  column.    Note  as  an  important 

4270.864062      special  case  of  our  rule:  //  of  two 


ON  THE  PRESENTATION  OF  ARITHMETIC.  123 

figures  multiplied  one  is  in  units'  column,  the  figure  put 
down  stands  under  the  other. 

There  are  two  interpretations  of  division,  namely 
.  .  .  Remainder  Division  and  Multiplication's  In 

verse.  Remainder  division  may  be  taught 
before  the  multiplication  of  fractions.  It  is  to  find  how 
many  times  one  number,  the  divisor,  is  contained  in  an- 
other, the  dividend;  and  what  then  remains.  For  ex- 
ample, if  eggs  are  four  cents  apiece,  how  many  can  be 
bought  for  three  nickels?  Answer  three  Or  in  count- 
ing with  a  compound  unit,  the  divisor,  how  many  times 
is  it  taken  before  overstepping  the  dividend? 

Historically  it  was  in  connection  with  measurement 
that  fractions  had  their  origin.  By  way  of  review  and 
advance  combined,  we  may  now  introduce  subtraction, 
of  course  never  to  be  worked  by  anything  but  addition, 
the  "making  change"  method. 

Again  multiplication  may  now  be  introduced,  with  the 
tables  for  doubling,  tripling,  quadrupling.  Here  may  be 
given  the  symbols,  +,  -,  x,  /,  -T-,  =. 

Pairs  of  numbers  may  now  be  exhibited  for  the 
child  to  give  their  difference;  then  pairs  of  numbers, 
the  second  number  a  1,  2,  3,  4,  or  5,  for  the  child  to 
give  the  product.  For  games  we  have  dominoes,  bean 
matching,  and  the  like.  Use  the  savage  device  of  a  row 
of  men  for  counting,  to  make  easy  our  positional  nota- 
tion for  number.  Thus  familiarize  digits  of  different 
orders. 

Sticks  and  stick-bundles  can  be  correlated  with  cents, 
nickels,  dollars,  halves,  quarters.  If  the  sticks  be  marked 
off  in  tenths,  decimals  may  be  illustrated. 

Thus  numbers  of  two  and  three  orders  are  familiar- 
ized, as  also  the  shifting  of  the  decimal  point.  9876  mills 


124       FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

are  987.6  cents,  or  98.76  dimes,  or  9.876  dollars.  Deci- 
mals and  fractions  are  made  simple  by  the  idea  of  a  prin- 
cipal unit  and  subunits. 

Give  each  child  a  cheap  foot  rule;  here  inches  are 
subunits.  Actual  familiarity  with  standards  for  measure 
is  essential,  the  more  so  as  these  are  no  part  of  pure  arith- 
metic or  number,  but  only  extraneous  components  of  a 
device  for  the  application  of  number,  namely  measure- 
ment. 

Practice  in  simple  multiplication,  envisaged  first  as 
condensed  addition,  may  go  up  through  doubling,  tripling, 
quadrupling,  quintupling.  Multiplication  by  ten  is  equiv- 
alent to  shifting  the  decimal  point  to  the  right.  Quin- 
tupling is  shifting  the  point  and  halving.  Measurements 
for  the  application  of  number  knowledge  to  the  attain- 
ment of  ends  desired  by  the  child  are  in  place,  but  the 
so-called  "formal  work"  and  "mechanical  drill"  may  give 
more  joy  and  interest  to  the  child  than  any  measurement. 

From  Teachers  College  Record  we  quote :  "Upon  be- 
ing given  their  choice  one  morning  between  going  to  the 
new  gymnasium  and  remaining  in  the  room  to  learn  a 
new  multiplication  table,  all  but  three  of  a  class  of  thirty 
chose  the  mental  gymnastics.  This  is  cited  to  show  that 
much  of  the  so-called  'formal  work,'  'systematic  me- 
chanical drill,'  which  sounds  so  formidable  to  an  out- 
sider, may  bring  much  delight  to  one  of  our  eight  year 
old  children,  and  that  the  mechanism  of  number  may  be 
secured  with  no  sacrifice  of  interest." 

Summary  (Second  Grade) 

A.  The  extra-arithmetical   idea  of  a  standard   for 
measurement. 

B.  The  usual  standards  for  measure. 


ON  THE  PRESENTATION  OF  ARITHMETIC.  125 

C.  Explanation  of  "to  measure." 

D.  Knowledge  obtained  by  measuring  is  a  combine 
of  number-knowledge  and  knowledge  of  the  standard. 

E.  Length,  area,  volume,  capacity,  weight,  tempera- 
ture;  with   their  standards,    foot,    square,   cube,   quart, 
pound,  degree. 

F.  Metric  description  evoking  visual  image. 

G.  Positional  notation  for  number. 

H.  Decimals.  Basal  subunits.  Significant  figures  to 
right  of  units  column. 

I.  Fractions.  Any  subunits. 

J.  Subtraction.     Difference. 

K.  Multiplication. 

L.  Symbols. 

M.  Games. 

O.  Change  of  unit.     Shifting  the  decimal  point. 

P.  Problems;  written  work. 

Q.  Multiplication  tables  through  quintupling. 

THIRD  GRADE. 

We  are  more  than  ever  to  aim  at  helping  the  develop- 
ment of  the  child  in  mental  power,  accuracy,  and  pre- 
cision, mind-mastery,  ability  to  direct  and  fix  the  atten- 
tion, and  withal  to  a  distinct  growth  in  technically  arith- 
metical equipment  for  efficiency  and  life. 

There  very  often  seems  here  to  bloom  out  spontane- 
ously in  the  child  a  love  for  what  has  sometimes  been 
called  the  abstract  formal  part  of  arithmetic.  It  is  seen 
to  give  delight.  The  play-joy,  which  is  perhaps  a  greater 
ingredient  in  pure  science  than  has  been  suspected,  now 
shows  forth  to  illumine  the  work,  and  beautify  the  seem- 
ingly mechanical. 


126        FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

Review  Work. 

A.  Counting  with  a  compound  unit,  by  2's,  by  3's, 
by  4's,  by  5's,  by  10's.     Beginning  with  zero  or  any 
number. 

B.  Addition  with  "carrying." 

C.  Subtraction,  with  "carrying."      (Never  use  any 
but  the  addition  method.) 

Advance  Work. 

D.  Multiplication.     Complete  the  tables  through  8 
constructively.    Explain  the  nine,  ten,  and  eleven  tables, 
so  that  they  need  not  be  memorized.     For  example,  to 
"nine-times"  a  digit,  write  the  preceding  digit  and  adjoin 
what  it  lacks  of  being  nine :  e.  g.,  9x8  =  72.    Connect  the 
eights  with  the   fours.     Written  multiplication;   begin 
always  with  the  left-most  figure  of  the  multiplier. 

E.  Division.     Two  kinds,  but  teach  first  Remainder 
Division.     First  utilize  the  multiplication  work.     Teach 
to  divide  by  one  digit,  then  by  two.    Contrast  remainder 
division  and  multiplication's  inverse. 

F.  Decimals.    The  point  in  addition  and  subtraction. 
Shifting  the  point  in  multiplication  and  division. 

G.  Fractions.     1/2,  1/4,  1/8,  1/3,  1/6;  change  the 
subunit.     Addition;  subtraction. 

H.  Measurement.     Square  measure. 

When  objects  are  used,  it  should  be  remembered  that 
after  they  have  once  served  their  purpose  they  only  ham- 
per children  and  teacher.  But  buying,  selling,  making 
change  may  often  be  used.  Let  the  children,  where  pos- 
sible, make  their  own  problems.  Groups  of  objects  may 
be  used  to  introduce  division.  Let  a  child  realize  what 
he  is  working  to  accomplish. 


ON  THE  PRESENTATION  OF  ARITHMETIC.  127 


FOURTH  GRADE. 

A.  A  review  of  addition,  subtraction,  and  multiplica- 
tion; but  a  very  extensive  presentation  and  mastery  of 
remainder  division. 

B.  Verifications.    Verify  addition  and  subtraction  by 
the  commutative   principle.      Verify  multiplication  and 
division  by  the  simplest  method  of  casting  out  nines. 

C.  Multiplication's  inverse.    No  remainder.  Fraction 
in  quotient. 

D.  Invention  of  problems. 

E.  Tests  of  accuracy  and  speed. 

F.  Measurements.    Include  decimals  and  fractions  in 
the  problems  apart  and  together.     Cubic. 

G.  Plotting  on  squared  paper.     Graphic  representa- 
tion. 

H.  Illustrations  of  the  life-value  of  facility  and  ac- 
curacy in  the  four  operations. 

I.  Divisibility.     Factors.     Multiples. 

K.  Emphasize  the  form  of  arrangement  of  written 
work. 

FIFTH  GRADE. 

A.  Decimals.    The  identity  of  decimal  notation  with 
the  ordinary  positional  notation  used  throughout  the  first 
four  grades. 

B.  Reading  of  all  decimals  in  the  new  method. 

C.  Addition  and  subtraction  shown  to  involve  nothing 
new. 

D.  Illustrations  from  our  money. 

E.  Multiplication  of   decimals;    (all   multiplications 
begin  with  the  left-most  figure  of  the  multiplier).    Shift- 
ing the  point. 


128        FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

F.  Division  of  decimals.     Shifting  the  point. 

G.  Problems. 

H.  Checking  results. 

I.  Percentage.  Applications  to  discount,  commission, 
simple  interest.  The  one  hundred  months  method  for 
interest. 

1.  Find  a  percent  of  a  number,   (given  number 

and  rate). 

2.  Find  what  percent  one  number  is  of  another. 

3.  To  find  a  number  from  a  given  percent  of  it. 
J.  Geometric  forms.     Denominate  numbers. 

K.  Prime  numbers.    Prime  factors. 
L.  Business  problems. 

SIXTH  GRADE. 
Fractions. 

Meaning  of  fractions.     Gain  by  the  notation. 

A.  Reduction,  that  is,  change  of  the  subunit. 

B.  Addition  and  subtraction.     Meaning  of  these  ope- 
rations for  fractions. 

C.  Least  common  multiple.     Common  denominator. 
Simplest  form. 

D.  Extension  of  the  idea  of  multiplication. 

E.  Multiplication  by  a  fraction. 

F.  "Of"  not  multiplication  symbol,  yet  %xQ  or  % 
times  Q  equals  %  of  Q. 

G.  Cancellation. 

H.  Division  by  a  fraction. 

I.  The  so-called  business  fractions  and  their  percent 
equivalents. 

J.  Expression  of  decimals  as  fractions  and  fractions 


ON  THE  PRESENTATION  OF  ARITHMETIC.  129 

as  decimals.     Show  by  squared  paper  and  diagrams  the 
identity  of  different  expressions  for  the  same  fraction. 
K.  Scale  drawing. 

SEVENTH  GRADE. 

Review. 

A.  Symbols:  Row  of  savages.  Zero.  Decimal  point. 
Fracjonal  notation.  Parentheses.  Units  added  counted 
together  are  thereby  taken  as  equivalent.  Illustrations. 
Adding  with  time  limit. 

B.  Business  forms  and  operations.    Banks.    Interest. 
Deposit    slips.      Checks.      Drafts.      Notes.      Discount. 
Stocks.     Bonds.     Coupons. 

C.  Meaning  of  per  cent  and  percentage.     Decimals 
and  fractions  in  percentage. 

D.  Percentage  equivalents  of  1/2,  1/3,  2/3,  1/4,  3/4, 
1/5,  1/6,  1/8,  3/8,  5/8,  7/8,  when  considered  as  ope- 
rators; and  vice  versa.     Percentage  equivalents  of  deci- 
mals when  considered  as  operators. 

E.  Problems  on  percentage.  Commission,  taxes  tariff, 
insurance. 

F.  Longitude  and  time. 

G.  Hundred  Months  Method. 

Interest  for  one  hundred  months  at  twelve  percent 
equals  principal.  Interest  for  one  month  at  twelve  per- 
cent equals  .01  of  principal.  Interest  for  a  number  of 
months,  an  aliquot  part  of  one  hundred,  is  just  that  part 
of  the  principal.  Interest  for  3  days  is  .  001  of  the  prin- 
cipal. 

Thus  to  get  interest  at  twelve  percent  for  eight 
months,  shift  point  two  places  to  left  in  principal  and 
multiply  by  eight. 


130        FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 

Interest  at  8,  6,  4,  3,  2  %  is  2/3,  1/2,  1/3,  1/4,  1/6 
of  that  at  12. 

H.  Mensuration ;  rectangle ;  parallelogram ;  trapezoid ; 
regular  polygon;  circle;  prismatoid;  Halsted's  Formula: 
V=  (a/4)  (B  +  3C) ;  prism;  cylinder;  pyramid;  cone; 
sphere. 

I.  Evolution;  use  of  tables.  Logarithms.  Negative 
and  positive  numbers.  The  equation.  The  unknown. 
The  variable.  The  constant.  The  parameter.  Coordi- 
nates. The  graph.  The  function. 

J.  General  review. 


INDEX. 


abacus  12,   17 

addition  29,  33,  44,  60,  1 16 

Ahmes  55 

angle  75 

Archimedes  76 

area  77 

arithmetic  68,  73,   101 

artificial  4 

associative  32 

assumptions   72,    102 

base  7,  24,  65 
Bayley  20 
begin  113 
binary   13 
Birch   $5 
blunders   no 
Bosworth  94 
Britannica  98 
Brown   97 
Byrhtferth  97 

calculus  22,   1 01 

cardinal  5,  in,  113 

cardinals  8 

carrying  121 

Cassini  99 

Century  94 

Chambers  96 

child  4,   no 

Chrysippos  6 

cipher  20 

columns  122 

commutative  31 

correlation  10 

count  n,  68,   m 

countable  84 

counting   10,   71,  88,   117 

Cowper  97 

cross-cut  79 


cuboid  79 
Cursor  97 

decimal  14,  51,   120 

decimals  22,  49,  63 

degree  75,  76 

difference  40,  122 

digits  22 

distributive  3t 

division  41,  47,  58,  61,  123 

dozen  97 

Egerton  21 
equivalent  10 
Eskimo  12 

fingers  n 
five   n 
formulas  32 
fraction   106 
fractions  56,  63,  121 

games  113 
geometry  75 
Gerbier  97 
Girard  28 
grades  no,  118 

Hale  97 

Halsted's  79,  80,  130 
Hamilton  32 
Hankel  22,   56 
Harriot  27 
Hickes  96 
Hill  20 
Hindu  19 
Holmes  97 
Hooke  97 
hundred  94,  129 
hundredths  50 


132        FOUNDATION  AND  TECHNIC  OF  ARITHMETIC. 


individual  3 
induction  34 
inequality  27 
infinite  82 
integer  24 
interest   129 
intrinsic  23 
invariance  15 
inverse  39 

Lagrange  i,  99 
Langland  98 
length  76 
Leonard  21 
local  23 

Maurolycus  34 

Mazarin  99 

measurement  68,   73,   119 

million  98 

modulus  22 

Moore  101 

multiple  42 

multiplication  35,  38,   45,  61 

Murray  95,  97 

Napier  49 

Napoleon  99 

natural  24 

Nau  19 

Nemorarius  21 

nines  46 

nine-times  126 

nominal  92 

notation  26 

number  5,  14,  69,  88,  92 

numbers  3 

numeral    12 

numeration  15,  23 

one  6 

order  81,  83 

ordered    59 

ordinal  25,  33,  88,   in 

ordinals  Si,  89,  no 

Oughtred  35 

parentheses  28 
part  28 

partitioned    14 
Peacock  56,  96 
periodicity    13 
permanence  56 
Planudes  21 


pi  ay- joy  125 
plus  29,  53 
point  51 
position  17 
positional  22 
prehuman  3 
presentation  no 
prism  79 
prismatoid  78 
product  35,   52 
Ptolemy   19 

quartets  117 
quotient  41,  53 

radian  77 
Raleigh  27 
ray  75 
read  51,  94 
reciprocal  58 
Recorde  26 
recur  64 
remainder  41 
roundheads  93 

Sacrabosco  21 

scale  59 

schoolmaster's  99 

Sebokt   19 

sect  75 

sense  82 

Servois  31 

seven  9 

Shakespeare  98 

shift  50 

solidus  43 

sphere  80 

standard   n,  73,   124 

Stevinus  49 

straight  75 

substitution  29 

subtraction  39,  44,  57,  116 

sum  30 

summits  78 

symbol  26 

symbols  18 

symmetrical  43 

teaching  23 
technic  44 
telephone  92 
ten  7 
tenths  50 


INDEX.  133 


terms  30  verify  46,  48 

thousand  96,  98  Vieta  31 

thousandths  51  volume  78,  80 
three  9,   116 

twenty  22  well-ordered  86 

two,  6,  9,   116  Whitney  99 

Widman  29 
unification  4 

"nit,  8,  73  zero  20 


THE  OPEN  COURT  MATHEMATICAL  SERIES 


A  Brief  History  of  Mathematics. 

By  the  late  DR.  KARL  FINK,  Tubingen,  Germany.  Trans- 
lated by  Wooster  Woodruff  Beman,  Professor  of  Math- 
ematics in  the  University  of  Michigan,  and  David  Eugene 
Smith,  Professor  of  Mathematics  in  Teachers'  College, 
Columbia  University,  New  York  City.  With  ^biographical 
notes  and  full  index.  Second  revised  edition.  Pages, 
xii,  333.  Cloth,  $1.50  net.  (5s.  6d.  net.) 

"Dr.  Fink's  work  Is  the  most  systematic  attempt  yet  made  to  present  a 
compendious   history  of  mathematics." — The  Outlook. 
"This  book  Is  the  best  that  has  appeared  In  English.     It  should  find  a 
place  In  the  library  of  every  teacher  of  mathematics." 

— The  Inland  Educator. 

Lectures  on  Elementary  Mathematics. 

By  JOSEPH  Louis  LAGRANGE.  With  portrait  and  biography 
of  Lagrange.  Translated  from  the  French  by  T.  J.  Mc- 
Cormack.  Pages,  172.  Cloth,  $1.00  net  (4s.  6d.  net.) 

"Historical  and  methodological  remarks  abound,  and  are  so  woven  to- 
gether •with  the  mathematical  material  proper,  and  the  whole  is  so 
vivified  by  the  clear  and  almost  chatty  style  of  the  author  as  to  give 
the  lectures  a  charm  for  the  readers  not  often  to  be  found  in  mathe- 
matical works." — Bulletin  American  Mathematical  Society. 

A  Scrapbook  of  Elementary  Mathematics. 

By  WM.  F.  WHITE,  State  Normal  School,  New  Paltz,  N. 
Y.  Cloth.  Pages,  248.  $1.00  net.  (5s.  net) 
A  collection  of  Accounts,  Essays,  Recreations  and  Notes, 
selected  for  their  conspicuous  interest  from  the  domain  of 
mathematics,  and  calculated  to  reveal  that  domain  as  a 
world  in  which  invention  and  imagination  are  prodigiously 
enabled,  and  in  which  the  practice  of  generalization  is  car- 
ried to  extents  undreamed  of  by  the  ordinary  thinker,  who 
has  at  his  command  only  the  resources  of  ordinary  lan- 
guage. A  few  of  the  seventy  sections  of  this  attractive 
book  have  the  following  suggestive  titles :  Familiar  Tricks, 
Algebraic  Fallacies,  Geometric  Puzzles,  Linkages,  A  Few 
Surprising  Facts,  Labyrinths,  The  Nature  of  Mathematical 
Reasoning,  Alice  in  the  Wonderland  of  Mathematics.  The 
book  is  supplied  with  Bibliographic  Notes,  Bibliographic 
Index  and  a  copious  General  Index. 

"The  book  Is  interesting,  valuable  and  suggestive.  It  Is  a  book  that 
really  fills  a  long-felt  want.  It  is  a  book  that  should  be  In  the  library 
of  every  high  school  and  on  the  desk  of  every  teacher  of  mathematics." 

— >T/ie  Educator-Journal, 


THE  OPEN  COURT  MATHEMATICAL  SERIES 

Essays  on  Mathematics. 

Articles  by  HENRI  POINCARE.     Published  in  the 
Monist.     Price,  60  cents  each. 

On  the  Foundations  of  Geometry Oct. ,  1898 

The  Principles  of  Mathematical  Physics Jan.,  1905 

Relations  Between   Experimental    Physics   and 

Mathematical  Physics July,  1902 

The  Choice  of  Facts April,  1909 

The  Future  of  Mathematics Jan.,  1910 

Mathematical  Creations July,  1910 

Chance Jan. ,  1912 

The  New  Logics April,  1912 

Portraits  of  Eminent  Mathematicians. 

Three  portfolios  edited  by  DAVID  EUGENE  SMITH, 
Ph.  D.,  Professor  of  Mathematics  in  Teachers' College, 
Columbia  University,  New  York  City. 

Accompanying  each  portrait  is  a  brief  biographical 
sketch,  with  occasional  notes  of  interest  concerning 
the  artist  represented.  The  pictures  are  of  a  size  that 
allows  for  framing  11x14. 

Portfolio  No.  1.  Twelve  great  mathematicians  down  to 
1700  A.  D.:  Thales,  Pythagoras,  Euclid,  Archi- 
medes, Leonardo  of  Pisa,  Cardan,  Vieta,  Napier, 
Descartes,  Fermat,  Newton,  Leibnitz.  Price, 
per  set,  $3.00.  Japanese  paper  edition,  $5.00. 

Portfolio  No.  2.  The  most  eminent  founders  and  pro- 
moters of  the  infinitesimal  calculus:  Cavallieri, 
Johann  &  Jakob  Bernoulli,  Pascal,  L'Hopital, 
Barrow,  Laplace,  Lagrange,  Euler,  Gauss,  Monge, 
and  Niccolo  Tartaglia.  Price,  per  set,  $3.00. 
Japanese  paper  edition,  $5.00. 

Portfolio  No.  3.  Eight  portraits  selected  from  the  two 
former  portfolios,  especially  adapted  for  high 
schools  and  academies.  Price,  $2.00.  Japan 
vellum,  $3.50.  Single  portraits,  35c.  Japan 
vellum,  50c. 


THE  OPEN  COURT  MATHEMATICAL  SERIES 


Essays  on  the  Theory  of  Numbers. 

(1)  Continuity  and  Irrational  Numbers,  (2)  The  Nature 
and  Meaning  of  Numbers.  By  RICHARD  DEDEKIND.  From 
the  German  by  W.  W.  BEMAN.  Pages,  115.  Cloth,  75 
cents  net.  (3s.  6d.  net.) 

These  essays  mark  one  of  the  distinct  stages  in  the  devel- 
opment of  the  theory  of  numbers.  They  give  the  founda- 
tion upon  which  the  whole  science  of  numbers  may  be  es- 
tablished. The  first  can  be  read  without  any  technical, 
philosophical  or  mathematical  knowledge;  the  second  re- 
quires more  power  of  abstraction  for  its  perusal,  but  power 
of  a  logical  nature  only. 

"A  model  of  clear  and  beautiful  reasoning." 

— Journal  of  Physical  Chemistry. 

"The  work  of  Dedeklnd  Is  very  fundamental,  and  I  am  glad  to  have  it 
in  this  carefully  wrought  English  version.  I  think  the  book  should  be 
of  much  service  to  American  mathematicians  and  teachers." 

— Prof.  E,  H.  Moore,  University  of  Chicago. 

"It  is  to  be  hoped  that  the  translation  will  make  the  essays  better 
known  to  English  mathematicians  ;  they  are  of  the  very  first  importance, 
and  rank  with  the  work  of  Weierstrass,  Kronecker,  and  Cantor  in  the 
same  field." — Nature. 


Elementary  Illustrations  of  the  Differential 
and  Integral  Calculus. 

By  AUGUSTUS  DE  MORGAN.  New  reprint  edition.  With 
subheadings  and  bibliography  of  English  and  foreign  works 
on  the  Calculus.  Price,  cloth,  $1.00  net.  (4s.  6d  net.) 

"It  aims  not  at  helping  students  to  cram  for  examinations,  but  to  give 
a  scientific  explanation  of  the  rationale  of  these  branches  of  mathe- 
matics. Uke  all  that  De  Morgan  wrote,  it  is  accurate,  clear  and 
philosophic." — Literary  World,  London. 


On   the    Study  and   Difficulties  of  Mathe- 
matics. 

By  AUGUSTUS  DE  MORGAN.  With  portrait  of  De  Morgan, 
Index,  and  Bibliographies  of  Modern  Works  on  Algebra, 
the  Philosophy  of  Mathematics,  Pangeometry,  etc.  Pages, 
viii,  288.  Cloth,  $1.25  net.  (5s.  net.) 

"The  point  of  view  is  unusual ;  we  are  confronted  by  a  genius,  who, 
like  his  kind,  shows  little  heed  for  customary  conventions.  The  'shak- 
ing up'  which  this  little  work  will  give  to  the  young  teacher,  the  stim- 
ulus and  implied  criticism  it  can  furnish  to  the  more  experienced,  make 
its  possession  most  desirable." — Michigan  Alumnus. 


THE  OPEN  COURT  MATHEMATICAL  SERIES 


The  Foundations  of  Geometry. 

By  DAVID  HILBERT,  Ph.  D.,  Professor  of  Mathematics  in 
the  University  of  Gottingen.  With  many  new  additions 
still  unpublished  in  German.  Translated  by  E.  J.  TOWN- 
SEND,  Ph.  D.,  Associate  Professor  of  Mathematics  in  the 
University  of  Illinois.  Pages,  viii,  132.  Cloth,  $1.00  net 
(4s.  6d  net.) 

"Professor  Hilbert  has  become  so  well  known  to  the  mathematical 
•world  by  his  writings  that  the  treatment  of  any  topic  by  him  commands 
the  attention  of  mathematicians  everywhere.  The  teachers  of  elemen- 
tary geometry  in  this  country  are  to  be  congratulated  that  it  is  possible 
for  them  to  obtain  in  English  such  an  important  discussion  of  these 
points  by  such  an  authority." — Journal  of  Pedagogy. 

Euclid's  Parallel  Postulate :  Its  Nature, Val- 
idity and  Place  in  Geometrical  Systems. 

By  JOHN  WILLIAM  WITHERS,  Ph.  D.  Pages  vii,  192.  Cloth, 
net  $1.25.  (4s.  6d.  net.) 

"This  is  a  philosophical  thesis,  by  a  writer  who  Is  really  familiar  with 
the  subject  on  non-Euclidean  geometry,  and  as  such  it  is  well  worth 
reading.  The  first  three  chapters  are  historical ;  the  remaining  three 
deal  with  the  psychological  and  metaphysical  aspects  of  the  problem ; 
finally  there  is  a  bibliography  of  fifteen  pages.  Mr.  Withers's  critique, 
on  the  whole,  is  quite  sound,  although  there  are  a  few  passages  either 
vague  or  disputable.  Mr.  Withers's  main  contention  is  that  Euclid's 
parallel  postulate  is  empirical,  and  this  may  be  admitted  in  the  sense 
that  his  argument  requires ;  at  any  rate,  he  shows  the  absurdity  of 
some  statements  of  the  a  priori  school." — Nature. 

Mathematical  Essays  and  Recreations* 

By  HERMANN   SCHUBERT,  Professor  of  Mathematics   in 

Hamburg.    Contents:    Notion  and  Definition  of  Number; 

Monism  in  Arithmetic;   On  the  Nature  of  Mathematical 

Knowledge;  The  Magic  Square;  The  Fourth  Dimension; 

The  Squaring  of  the  Circle.    From  the  German  by  T.  J. 

McCormack.     Pages,  149.     Cuts,  37.    Cloth,  75  cents  net. 

(3s.  6d.  net.) 

"Professor  Schubert's  essays  make  delightful  as  well  as  Instructive 
reading.  They  deal,  not  with  the  dry  side  of  mathematics,  but  with  the 
philosophical  side  of  that  science  on  the  one  band  and  its  romantic  and 
mystical  side  on  the  other.  No  great  amount  of  mathematical  knowl- 
edge is  necessary  in  order  to  thoroughly  appreciate  and  enjoy  them. 
They  are  admirably  lucid  and  simple  and  answer  questions  in  which 
every  intelligent  man  is  interested." — Chicago  Evening  Post. 
"They  should  delight  the  jaded  teacher  of  elementary  arithmetic,  who 
Is  too  liable  to  drop  into  a  mere  rule  of  thumb  system  and  forget  tho 
scientific  side  of  his  work.  Their  chief  merit  is  however  their  intel- 
ligibility. Even  the  lay  mind  can  understand  and  take  a  deep  interest 
in  what  the  German  professor  has  to  say  on  the  history  of  magic 
squares,  the  fourth  dimension  and  squaring  of  the  circle." 

— Saturday  Review. 


THE  OPEN  COURT  MATHEMATICAL  SERIES 

On  the  Foundation  and   Technic   of  Arith- 
metic. 

By  GEORGE   BRUCE  HALSTED.     Cloth,   $1.50 
Pages,  140. 

A  practical  presentation  of  arithmetic  for  the  use  of 
teachers.  There  has  been  in  mathematics  an  outburst 
of  unexpected  deep  reaching  progress  and  properly  to 
understand  or  to  teach  arithmetic,  one  should  have  a 
glimpse  of  its  origin,  foundation,  meaning  and  aim. 

Non-Euclidean    Geometry,    a    Critical    and 
Historical  Study  of  its  Development. 

By  ROBERTO  BONOLA.  With  an  Introduction  by 
FEDERIGO  ENRIQUES.  Translated  by  H,  S. 
CARSLAW.  Cloth,  $2.00.  Pages,  268.  Illus- 
trated. 

A  clear  exposition  of  the  principles  of  elementary 
geometry  especially  of  that  hypothesis  on  which  rests 
Euclid's  theory  of  parallels,  and  of  the  long  discussion 
to  which  that  theory  was  subjected;  and  of  the  final 
discovery  of  the  logical  possibility  of  the  different 
Non-Euclidean  Geometries. 


In  Preparation:  Bibliography  of  1OO 
selected  books  on  the  History  and  Phi- 
losophy of  Mathematics. 

Price,  $1.00. 


THE  OPEN  COURT  MATHEMATICAL  SERIES 


Geometric  Exercises  in  Paper-Folding. 

By  T.  SUNDARA  Row.  Edited  and  revised  by  W.  W.  BE- 
MAN  and  D.  E.  SMITH.  With  half-tone  engravings  from 
photographs  of  actual  exercises,  and  a  package  of  papers 
for  folding.  Pages,  x,  148.  Price,  cloth,  $1.00  net.  (4s. 
6d.  net.) 

"The  book  is  simply  a  revelation  In  paper  folding.  All  sorts  of  things 
are  done  with  the  paper  squares,  and  a  large  number  of  geometric 
figures  are  constructed  and  explained  in  the  simplest  way." 

— Teachers'  Institute. 

Magic  Squares  and  Cubes. 

By  W.  S.  ANDREWS.  With  chapters  by  PAUL  CARUS,  L.  S. 
FRIERSON  and  C.  A.  BROWNE,  JR.,  and  Introduction  by 
PAUL  CARUS.  Price,  $1.50  net.  (7s.  6d.  net.) 
The  first  two  chapters  consist  of  a  general  discussion  of  the 
general  qualities  and  characteristics  of  odd  and  even  magic 
squares  and  cubes,  and  notes  on  their  construction.  The 
third  describes  the  squares  of  Benjamin  Franklin  and  their 
characteristics,  while  Dr.  Carus  adds  a  further  analysis 
of  these  squares.  The  fourth  chapter  contains  "Reflections 
on  Magic  Squares"  by  Dr.  Carus,  in  which  he  brings  out 
the  intrinsic  harmony  and  symmetry  which  exists  in  the 
laws  governing  the  construction  of  these  apparently  mag- 
ical groups  of  numbers.  Mr.  Frierson's  "Mathematical 
Study  of  Magic  Squares,"  which  forms  the  fifth  chapter, 
states  the  laws  in  algebraic  formulas.  Mr.  Browne  con- 
tributes a  chapter  on  "Magic  Squares  and  Pythagorean 
Numbers,"  in  which  he  shows  the  importance  laid  by  the 
ancients  on  strange  and  mystical  combinations  of  figures. 
The  book  closes  with  three  chapters  of  generalizations  in 
which  Mr.  Andrews  discusses  "Some  Curious  Magic 
Squares  and  Combinations,"  "Notes  on  Various  Con- 
structive Plans  by  Which  Magic  Squares  May  Be  Classi- 
fied," and  "The  Mathematical  Value  of  Magic  Squares." 

"The  examples  are  numerous ;  the  laws  and  rules,  some  of  them 
original,  for  making  squares  are  well  worked  out.  The  volume  is 
attractive  in  appearance,  and  what  is  of  the  greatest  importance  in 
such  a  work,  the  proof-reading  has  been  careful." — The  Nation. 

The  Foundations  oi  Mathematics. 

A  Contribution  to  The  Philosophy  of  Geometry.  BY  DR. 
PAUL  CARUS.  140  pages.  Cloth.  Gilt  top.  75  cents  net. 
(3s.  6d.  net.) 


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